Reaction Graphs for Some Complete Graph Decompositions
Zuzana Masárová
supervisor: Alexander Rosa
Math 790
June 2012
Acknowledgements
I’d like to thank to Prof. Alexander Rosa for choosing an interesting topic in graph theory, taking me
as his student and being a great supervisor throughout the year.
To Prof. Deirdre Haskell and Prof. Manfred Kolster I’d like to thank for their decision to admit me into
the program in an extremely short time, several months after the usual deadline.
Contents
1 Introduction
1.1 Reaction graphs in mathematical chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Mathematical variations of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Outline of the problem investigated in this project . . . . . . . . . . . . . . . . . . . . . . . .
4
4
5
5
2 Subgraphs for which K6 decomposes
2.1 Subgraphs A to I . . . . . . . . . .
2.2 Subgraphs J, K . . . . . . . . . . . .
2.3 Subgraph L . . . . . . . . . . . . . .
2.4 Subgraph M . . . . . . . . . . . . .
2.5 Subgraph N . . . . . . . . . . . . . .
2.6 Subgraph O . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
6
7
7
8
8
8
9
3 Number of vertices in corresponding
3.1 Subgraph A . . . . . . . . . . . . . .
3.2 Subgraph B . . . . . . . . . . . . . .
3.3 Subgraph C . . . . . . . . . . . . . .
3.4 Subgraph D . . . . . . . . . . . . . .
3.5 Subgraph E . . . . . . . . . . . . . .
3.6 Subgraph F . . . . . . . . . . . . . .
3.7 Subgraph G . . . . . . . . . . . . . .
3.8 Subgraph H . . . . . . . . . . . . . .
3.9 Subgraph I . . . . . . . . . . . . . .
reaction graphs
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
11
11
11
12
13
14
14
15
17
20
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4 Reaction graphs corresponding to subgraphs A, B and C
23
4.1 Reaction graphs: methods of construction, definitions of adjacency and investigated properties 23
4.1.1 Isomorphic systems, regular graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Reaction graph corresponding to subgraph A . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Reaction graphs corresponding to subgraph B . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.1 Adjacency defined by an overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3.2 Adjacency defined by a transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.4 Reaction graphs corresponding to subgraph C . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4.1 Adjacency defined by an overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4.2 Adjacency defined by a transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Conclusions and further study
34
3
1
Introduction
Reaction graphs originate in mathematical chemistry, where they have first been used to describe a special
kind of chemical reaction known as degenerate rearrangement. The concept has later been reformulated in
more abstract terms and explored by employing mathematical tools, especially the permutation group theory.
Variations of the problem including decompositions of a complete graph instead of molecular graphs and a
theoretically defined change instead of chemical rearrangements have subsequently also been developed and
studied by mathematicians.
This research project follows on the work of Lindner and Rosa in [5]. It investigates some kinds of
decompositions of K6 and several examples of resulting reaction graphs.
In this section I give an introduction and a short review of the history of reaction graphs and outline the
problem studied in this research project.
1.1
Reaction graphs in mathematical chemistry
Degenerate rearrangements and their reaction graphs have been studied in mathematical chemistry in
the past few decades. A rearrangement is a kind of chemical reaction “during which a chemical compound
transforms into an isomeric compound (that is, a compound with the same chemical formula)”, as explained
in [7]. If the resulting compound is isomorphic, we speak of a degenerate rearrangement. Moreover, Klin,
Tratch and Zefirov in [3] call a rearrangement as highly degenerate, if there are more than two isomorphic
states between which the compound is being transformed (in what follows this means that the reaction graph
has more than two vertices).
A reaction graph is a graph, which, loosely speaking, has as vertices all the possible states of the compound and whose edges represent the rearrangements between the respective states. The ultimate aim is
to understand the properties of the reaction graph, as these comprise much information about the reactions
involved.
Mathematically, the reaction graph is constructed as follows.
Firstly, the compound is represented by a molecular graph Γ. The molecular graph can be the actual
chemical formula of the compound, or just its part which is crucial in the rearrangement process.
Different numberings of the molecular graph Γ then correspond to different states of the compound. If
Γ has n vertices then there are n! possible numberings. The numberings which can be obtained from each
other by some kind of an automorphism of the molecular graph Γ (the type of the automorphism is usually
specified on chemical grounds) are considered to be the same. Thus, there are n!/|H| different states Γi of
the compound, where H is some subgroup of the automorphism group Aut(Γ) of Γ. This means that the
reaction graph has n!/|H| vertices Γi . The vertices can be viewed simply as equivalence classes of possible
numberings as in [7].
Secondly, the edges in the reaction graph are defined. An edge appears between the states Γi and Γj if
and only if some numbering in the equivalence class of the state Γi can be transformed by the rearrangement
process into a numbering in the equivalence class of Γj (cf. [7])1 . The edges of the reaction graph are usually
considered to be undirected, as the reaction can go in both directions.
Some properties of the reaction graph R which are of interest to mathematical chemists have been summarised by Klin and Zefirov in [4] and include the following:
• number of vertices in R and its connected components,
• connectivity criterion for R,
• automorphism group of R,
• diameter and girth of R,
• ways of efficient coding of R and
• visualisation of R.
1 Unlike
Lloyd and Jones in [7], Klin and Zefirov use in [4] an induced action of Sn on the set of numberings {Γi } to define
the edges of the reaction graph.
4
Many of the above properties have by now been investigated either in the cases of particular reaction
graphs or more systematically by using the tools from algebraic combinatorics or permutation group theory.
For example, Klin, Tratch & Zefirov give in [3] a connectivity criterion for a reaction graph R in the case when
R is a 2-orbit of the action of Sn on the set of different numberings {Γi }. For a review of other achievements
related to reaction graphs in mathematical chemistry see Klin and Zefirov in [4] and Lloyd and Jones in [7].
1.2
Mathematical variations of the problem
Mathematical variations of the original problem involving the reaction graphs have been studied by,
for example, Lloyd in [6] and Lindner and Rosa in [5]. Instead of a molecular graph Γ of a compound,
Lindner and Rosa take an edge-disjoint decomposition of the complete graph K6 into three copies of K4 − e
(a complete graph K4 with an edge removed). They call such a decomposition a system and find that there
are 30 distinct systems.
The systems form the vertices of the reaction graphs studied in [5]. Edges, which were previously determined by the rearrangement reaction are in this mathematical model defined by specifying a ‘small change’
between the pairs of systems. In [5], two systems are defined to be adjacent if they overlap on exactly i triangles, where i = 0, 2, 3 and each triangle K3 is a subgraph of some copy of K4 − e within the corresponding
system.
The resulting reaction graphs are denoted G(i) and their properties, such as the degree, the neighbourhood
graphs, hamiltonicity, the clique number, independence number, diameter and the order of the automorphism
group are investigated in the paper.
1.3
Outline of the problem investigated in this project
This project follows on the work done in [5]. It first investigates the ways in which a complete graph K6
can be decomposed into three isomorphic subgraphs; for each such decomposition it then finds the number
of vertices in a corresponding reaction graph and if this is reasonably small, a ‘small change’ is proposed to
define the edges in the reaction graph. The properties of the resulting reaction graphs are then studied.
The project splits naturally into three parts and so the discussion is divided in Sections 2, 3 and 4
accordingly. Firstly, in Section 2 we find all subgraphs S of K6 for which K6 can be decomposed into three
edge-disjoint copies isomorphic to S. It turns out that there exist nine such subgraphs.
For each of these nine subgraphs we subsequently calculate the number of vertices in a corresponding
reaction graph. Given a subgraph S, this amounts to finding all the different decompositions of K6 into
copies of S which are possible. This is done in Section 3. It turns out that four out of nine subgraphs
produce reaction graphs with 15, 30, 120 and 120 vertices, while the remaining five have each over 200
vertices in their reaction graphs. Only the four smallest reaction graphs are studied further.
The reaction graph with 30 vertices corresponds to a decomposition involving the subgraph K4 − e which
was already discussed in [5]. The other three subgraphs (with reaction graphs of order 15, 120 and 120) are
studied in Section 4. For each of these reaction graphs, the ‘small change’ determining the edges is chosen in
two ways:
1. as an overlap on significant edges, i. e. two systems are defined to be adjacent in the reaction graph if
and only if they overlap on some specified edges (possibly an empty set),
2. as a transposition, i. e. two systems are adjacent if and only if one can be obtained from the other by
a transposition of vertices.
In each reaction graph both types of the adjacency definition are compared and the properties such as
connectivity, number of neighbours and ways of visualisation of the resulting reaction graphs are studied.
5
2
Subgraphs for which K6 decomposes
The basic building blocks which we are interested in are the edge-disjoint decompositions of a complete
graph K6 into isomorphic subgraphs. There are 15 edges in K6 and so the decomposition is possible into
either three isomorphic subgraphs, each with 5 edges, or five subgraphs, each having 3 edges. In this project
we only investigate the former, the latter is left for further research.
All in all there are 15 graphs with 5 edges which can be drawn on 6 vertices. They are depicted below
(taken from [2]).
6
The claim is that the decomposition of K6 into three edge-disjoint isomorphic subgraphs is possible if
and only if the subgraphs are isomorphic to any one of the graphs A to I above. We prove this claim in the
remainder of the section.
2.1
Subgraphs A to I
The following diagrams show that if a subgraph S of K6 is isomorphic to any of the graphs A to I above,
then K6 can be decomposed into three edge-disjoint subgraphs isomorphic to S (the copies of S inside K6
are distinguished by colours).
2.2
Subgraphs J, K
If K6 were to be decomposed into three copies isomorphic to J (or K), then since J (and similarly K)
has no isolated vertices, each vertex of K6 would be incident to at least one edge from each copy . But this is
impossible, because each copy of J already contains a vertex with degree 5 (and K a vertex with degree 4).
7
2.3
Subgraph L
Any graph which consists of three edge-disjoint cycles contains only vertices with even degree. Since
vertices of K6 have degree 5, K6 cannot be decomposed into subgraphs isomorphic to L.
2.4
Subgraph M
Suppose that we want to build K6 from three (black, red and blue) graphs isomorphic to the graph M .
Then since in M there are only two vertices with odd degree and each vertex of K6 has degree 5, the black,
red and blue edges that join the vertices with odd degrees in their respective graphs must form a perfect
matching. Finishing off the black copy, the black 4-cycle will have one of the other edges from the matching
on the diagonal; wlog we can suppose that it is the red one. But then it is not possible to finish the red copy
so that it is isomorphic to M .
2.5
Subgraph N
Define the black, red and blue degree of a vertex v as the number of edges of the corresponding colour
which are incident to v in a decomposition of K6 .
8
Similarly as above, the graph N has only two vertices with odd degree. Therefore, in order to obtain
degree 5 everywhere, at each vertex only one of the black, red and blue degrees can be odd.
Suppose that the black copy has been fixed as in the figure above and we want to add the remaining two
copies. Consider the edge 15, we may assume that it is red. Since the vertices 1 and 5 already have an odd
black degree, their red degrees are even. Thus the edge 15 must be a part of a red 3-cycle. Then the rest
of the red copy is uniquely determined. However, this yields a blue copy which is not isomorphic to N , i. e.
the decomposition of K6 into subgraphs isomorphic to N is not possible.
2.6
Subgraph O
Graph O has one vertex of degree three, call it the main vertex. Call the edge joining the vertices which
are non-adjacent to the main vertex a tail edge.
We try to build K6 from three graphs isomorphic to O. First, we fix the black copy of O (as in the figure
above) and determine the red and blue main vertices. None of the vertices with the black degree equal to
two or three can be a main vertex for the red (or blue) copy.
Indeed, vertex 5 clearly cannot be the main vertex for two colours. Suppose that the vertex 4 is the main
red vertex. Then to avoid 3-cycles in the red copy, the 3-cycle 126 must be blue, contradiction.
Similarly, if the vertex 3 is the main red vertex, it uniquely determines the red tail edge, but then it is
impossible to add the blue main vertex with its edges and the blue tail edge.
9
Thus the two remaining main vertices are either the vertices 1 and 2 or 1 and 6. In both cases then pick
a colour of the edge joining these main vertices. Then the triple of edges incident to one of the main vertices
and the corresponding tail edge is uniquely determined, but it is not possible to finish the decomposition.
The procedure is very similar in all four cases which need to be investigated and so the following figures only
show one case (picking 1, 2 as main vertices and blue colour of the edge joining them).
10
3
Number of vertices in corresponding reaction graphs
Now that we know all the graphs for which K6 decomposes into three isomorphic edge-disjoint copies,
we find the number of vertices in corresponding reaction graphs. As explained in Section 1, given a subgraph
S, the vertices of a reaction graph are all possible edge-disjoint decompositions of K6 into three copies of S
(also called systems). In this section we calculate the number of different systems for each of the graphs A
to I.
Given a graph S, let k be the number of distinct labelled graphs on six vertices isomorphic to S, in other
words, k is the number of ways to draw a single copy of S on six (numbered) vertices. If n is the number of
different ways of finishing a decomposition once one copy of the subgraph has been fixed, then the number
of different systems is equal to kn
3 .
3.1
Subgraph A
!"
For the graph A on six vertices, there are 64 ways to choose the vertices of the 4-cycle and 3 ways to
!"
draw the 4-cycle on them. This uniquely determines the graph. Thus, k = 64 · 3 = 45.
The graph A has only two vertices of odd degree, therefore, by the same argument as in Section 2.4, the
black, red and blue edges that join the vertices with odd degree in their respective copies must form a perfect
matching. This matching is uniquely determined once one of the copies - without the loss of generality the
black one as in the figure - has been fixed. But then the position of the red and blue 4-cycles is uniquely
determined as well, finishing off the decomposition.
Therefore, n = 1. Hence, there are
subgraph A has 15 vertices.
3.2
45·1
3 ,
i. e. 15 different systems. A reaction graph corresponding to
Subgraph B
11
In the graph B, call the edge! joining
the vertices of degree 3 the main !edge.
"
" When fixing one
! " copy
! " of B,
the main edge can be chosen in 62 ways and the remaining edges then in 42 ways. Thus k = 62 · 42 = 90.
Since the endvertices of a main edge have degree three, no two main edges can share a vertex. The black,
red and blue main edges must therefore form a perfect matching. If we assume that the black copy has been
fixed as in the figure above, then the other main edges can be added in two ways: either as the edges 35 and
46, or 36 and 45 (having 34 and 56 as main edges is not possible, since both edges 23 and 24 are already
black and so we could not finish the red copy of B).
In both cases then pick a colour of an edge adjacent to both the red and blue main edges (in the two figures
above consider the colour of, for example, the edges 45 and 34 respectively). This gives us four possibilities
to investigate. It turns out that in each case the colours of the remaining edges in K6 are already uniquely
determined. The case of edges 35 and 46 being the main edges and a red colouring of the edge 45 is worked
out below, the other three cases are similar.
Thus, once one copy of B is fixed, there are 4 ways to finish the decomposition, and so n = 4. All in all
there are therefore 90·4
3 , i. e. 120 different systems and so a reaction graph corresponding to the graph B has
120 vertices.
3.3
Subgraph C
In the graph C, similarly as in Section 2.6, call the vertex with degree three the main vertex and the edge
joining the vertices non-adjacent to the main vertex an isolated edge. Notice that the main vertex with its
incident edges uniquely determines the isolated edge.
!"
When fixing a single copy of the graph C on six vertices, there are 62 ways to choose the isolated edge,
4 ways to further choose the vertices of the 3-cycle and, finally, 3 ways
! " to choose which vertex of the 3-cycle
has degree three. Thus, one copy of the graph C may be fixed in 62 · 4 · 3 ways, so k = 180.
12
Suppose now that the black copy of C has been fixed as in the figure above. We claim that the red and
blue main vertices can only be the vertices 1, 5 or 6. Indeed, 4 already is a black main vertex and so cannot
be the main vertex for a different colour. Similarly, if the vertices 2 or 3 were the main vertices, it would be
impossible to add the isolated edge of the corresponding colour, as the following figures show:
Thus, the remaining two main vertices can be either the vertices
Case 1.
1 and 5 or
Case 2.
5 and 6 or
Case 3.
1 and 6.
In Case 1 and 2 picking a colour of the edge joining the red and blue main vertex determines the colours
of all the remaining edges of K6 . However, in none of these cases we get a decomposition of K6 in which all
the subgraphs are isomorphic to the graph C.
Case 3 splits into four subcases according to the colour of the edges incident to the red main vertex.
In each subcase the red isolated edge, blue edges incident to the blue main vertex and the blue isolated
edge (in this order) are uniquely determined. Only the second subcase can then be finished (in two ways)
into a required decomposition of K6 , therefore n = 2.
Hence, there are 180·2
3 , i. e. 120 different systems corresponding to the graph C.
3.4
Subgraph D
13
Subgraph D is the graph K4 − e mentioned in Section 1 and already studied by Lindner and Rosa in [5].
They find that there are 30 systems corresponding to this subgraph and further investigate the properties of
a corresponding reaction graph. For details of all calculations see the above paper.
3.5
Subgraph E
!"
When fixing a single copy of the graph E on six vertices, there are 63 ways to choose the position of the
triangle, 3 ways to choose which
! " vertices of the triangle have degree three and 3 · 2 ways to pick the remaining
two edges. Therefore, k = 63 · 3 · 3 · 2 = 360.
In the graph E, call the edge joining the vertices with degree three the main edge. The black, red and
blue main edges must clearly form a perfect matching. If we assume that the black copy of E has been placed
as in the figure above, then the red and blue main edges can be chosen in three ways: either as edges 13 and
56 or as 36 and 15 or as 35 and 16.
In each case the colour of edges incident to vertex 3, then edges incident to vertices 5 and 1 (in one order
or another) and, finally, the edges incident to vertex 6 are uniquely determined. Only the first and third case
produce a decomposition of K6 in which all subgraphs are isomorphic to E. Thus, once one copy is fixed,
there are two different ways to finish a decomposition and n = 2.
Hence, there are 360·2
3 , i. e. 240 different systems in total corresponding to the graph E.
3.6
Subgraph F
In the graph F , call the !vertex
with degree four the main vertex. When fixing a single copy of the graph
"
F on six vertices, there are 63 ways to choose the position of the triangle, 3 ways to choose the main vertex
!"
in the triangle and 3 ways to draw the remaining two edges. Therefore, k = 63 · 3 · 3 = 180.
14
Suppose that the black copy has been fixed as in the figure above. The claim is that the remaining main
vertices can only be positioned in two ways: either at vertices 1 and 6 or at 5 and 6. Indeed, none of the
vertices 2, 3 or 4 can be a main vertex for the red or blue colour, since their black degree is more than one.
Also, if the remaining main vertices were 1 and 5, then the edge 15 would need to be both red and blue,
which is impossible.
Once the main vertices are chosen, their four incident edges of the corresponding colours are uniquely
determined. Both cases can then be finished in two different ways, as can be seen in the following figures.
.
Thus, given a fixed copy of the graph F , the decomposition can be finished in four ways, therefore n = 4.
Hence, there are 180·4
3 , i. e. 240 different systems. A reaction graph corresponding to the graph F has
240 vertices.
3.7
Subgraph G
!"
When fixing a single copy of the graph G on six vertices, there are 63 ways to choose the position of the
triangle, 3 ways to choose a vertex of the triangle with degree
! "three, and 3 · 2 ways to choose the remaining
two edges. Thus, one copy of the graph G can be drawn in 63 · 3 · 3 · 2 ways and k = 360.
In the graph G, call the vertices having degree of the corresponding colour zero, the black (or red or
blue) isolated vertices. Suppose that the black copy of the graph G has been fixed as in the figure above.
Then the vertices 5 and 6 can neither be red nor blue isolated vertices, as that would produce one copy in a
decomposition of K6 having a vertex of degree four, i. e. not being isomorphic to G. Also notice that the
red and blue isolated vertices must be chosen so that the edge joining them is black. Therefore, the red and
blue isolated vertices can either be
Case 1.
the vertices 1 and 2 or
Case 2.
the vertices 2 and 3 or
Case 3.
the vertices 2 and 4 or
Case 4.
the vertices 3 and 4.
15
Case 1.
In this case the colour of all edges incident to vertices 1 or 2 is determined, yielding a situation in the
following figure.
In order to make a blue 3-cycle, either the edge 35 or 45 must be blue. Choosing one or the other then
uniquely determines the rest of the blue copy. However, in neither case the red subgraph ends up being
isomorphic to G.
Case 2.
After colouring all edges incident to the red and blue isolated vertices 2 and 3 we get the following
situation.
In order to make a red 3-cycle, one of the edges 15 and 56 must be red. Also, to prevent a blue 4-cycle
from occuring, either the edge 46 or 45 must be red. To satisfy these conditions and to make the red copy
isomorphic to G, the remaining two red edges must be 15 and 46. This produces a blue copy isomorphic to
G and so a required decomposition of K6 .
Case 3.
Again, the colour of edges incident to the red and blue isolated vertices 2 and 4 is determined and
produces the following situation.
16
Similarly as above, to make a red 3-cycle, either the edge 15 or 56 must be red, and to prevent a blue
4-cycle, either the edge 35 or 36 must be red. The only possibility is to have red edges 15 and 36. The blue
copy is then isomorphic to G and so we get a second decomposition of K6 .
Case 4.
After colouring the edges incident to either the red or blue isolated vertex, we get the following.
To make a blue and a red 3-cycle, one of the edges 15 and 56 must be red and the other blue. The choice
of their colouring then uniquely determines the rest of the decomposition, however, in neither case produces
a decomposition of K6 in which all three subgraphs are isomorphic to G.
Summing up all the above cases, we have shown that if one copy of G is given, the decomposition can be
finished in two different ways and so n = 2.
Hence, there are 360·2
3 , i. e. 240 different systems corresponding to subgraph G.
3.8
Subgraph H
When fixing a single copy of the graph
H on six vertices, the vertex with degree three and its three
!"
adjacent vertices can be chosen in 6 · 53 ways. The remaining two edges can then be added in 3 · 2 ways.
!"
Therefore, one copy of H can be drawn in 6 · 53 · 3 · 2 ways and k = 360.
In the graph H, let the vertex with degree three be main vertex and suppose that the black copy of H
has been fixed as in the figure above. We claim that none of the vertices 1, 2 and 4 can be the red or blue
main vertex. This clearly holds for vertex 2, since 2 has black degree three. If we assume that vertex 1 is the
red (or blue) main vertex, then 1 must have blue (or red) degree zero, contradiction. Same argument holds
for vertex 4.
Thus, the red and blue main vertices can be vertices 3, 5 or 6. Considering also the colour of the edge
joining the two main vertices, we obtain six cases:
Case 1.
3 and 6 main vertices; edge 36 red
Case 2.
3 and 6 main vertices; edge 36 blue
Case 3.
3 and 5 main vertices; edge 35 red
17
Case 4.
3 and 5 main vertices; edge 35 blue
Case 5.
5 and 6 main vertices; edge 56 red
Case 6.
5 and 6 main vertices; edge 56 blue
In considering the above cases, the following facts about the decomposition of K6 into three copies of H
come useful:
i. The three edges of the same colour incident to a given main vertex v are determined once two edges of
another colour incident to this vertex are known.
ii. Black, red and blue degree of any given vertex v must be at least one.
iii. Given a colour c and a vertex v (different from the c main vertex), there is a path between v and the c
main vertex of colour c and length 1 or 2.
iv. In addition to iii, if the three edges of the same colour incident to the c main vertex are known, one of
their endvertices must be adjacent to v, by an edge with colour c.
In what follows, denote by ’iii (c, v)’ the fact iii applied to the colour c and vertex v. Similarly for the other
facts.
Case 1.
Case 1 assumes the situation in the figure above. Then, by i (6), the edges 26, 46 and 56 are blue. By
ii (2), the edge 25 is red. By iii (red, 2), the edge 35 is red. Then by iv (blue, 3) the edge 34 is blue. Finally,
by ii (4) the edge 14 is red and by iii (red, 4) the edge 13 is red, finishing the red copy and so also determining
the blue copy. We get a decomposition of K6 into three copies isomorphic to H.
Case 2.
Again, the situation assumed is in the figure above. By i (3), the edges 13, 35 and 34 are red. By i v(red,
2) the edge 25 is red. Finally, by iv (red, 6) and the fact that we want the red copy to be isomorphic to the
graph H, the edge 46 must be red, finishing the red copy. The blue subgraph is then uniquely determined,
however, not isomorphic to H.
18
Case 3.
The starting situation is in the figure above. By i (5), the edges 15, 25 and 56 are blue. By ii (2) the edge
26 is red. By iii(red, 2) the edge 36 is red. By iv (blue, 3) the edge 13 is blue. Finally, by iv (blue, 4) and the
fact that the blue copy needs to be isomorphic to H, the edge 46 is blue, finishing the blue copy. Red copy
is then uniquely determined, isomorphic to H and so we get another decomposition of K6 .
Case 4.
Again, the assumed situation is in the figure above. By i (3), the edges 13, 34 and 36 are red. By iv (red,
2), the edge 26 is red. Finally, by iv (red, 5) and the fact that we want the red copy to be isomorphic to H,
the edge 15 is red. However, the blue copy which is then uniquely determined is not isomorphic to H.
Case 5.
The starting situation is in the figure above. By i (6), the edges 26, 36 and 46 are blue. By ii (2), the edge
25 is red. By i v(blue, 5) the edge 35 is blue. Finally, by iv (blue, 1) and the fact that the blue copy needs
to be isomorphic to H, the edge 14 is blue, finishing the blue copy. However, the red copy does not end up
being isomorphic to H.
19
Case 6.
Again, the starting situation is above. By i (5), the edges 15, 25 and 35 are red. By ii (2), the edge 26
is blue. By iv (red, 6), the edge 36 is red. Finally, by iv (red, 4) and the fact that the red copy needs to be
isomorphic to H, the edge 14 is red. This finishes the red copy, however, also produces a blue copy which is
not isomorphic to H.
To sum up all the cases above, once one copy of H is fixed, there are two different ways to finish the
decomposition of K6 into three copies of H. Therefore, n = 2.
Hence, there are 360·2
3 , i. e. 240 different systems. A reaction graph corresponding to the graph H has
240 vertices.
3.9
Subgraph I
Six vertices can be ordered in 6! ways; always two orderings correspond to the same drawing of the graph
I. Therefore, graph I can be chosen in 6!
2 ways and k = 360.
In the graph I, call the vertices with degree one the endvertices. Since only the endvertices have an odd
degree in I, each vertex in a decomposition of K6 must be an endvertex for exactly one copy of I. Suppose
that the black copy has been fixed as in the figure above. Then, without the loss of generality, there are three
cases to consider:
Case 1.
2 and 3 are the red endvertices, 4 and 5 the blue endvertices
Case 2.
2 and 4 are the red endvertices, 3 and 5 the blue endvertices
Case 3.
2 and 5 are the red endvertices, 3 and 4 the blue endvertices
In considering the above cases, the following observations about a decomposition of K6 into three copies
of I come useful:
i. If a vertex v is an endvertex for colour c, then its c degree is one, otherwise the degree is two.
ii. Given a colour c, a path between the c endvertices in which all edges have colour c cannot have length
other than 5.
20
Case 1.
In this case we must have, by i (blue, 5) and i (red, 5), either the edge 35 blue and the edges 25 and 15
red, or the edge 25 blue and the edges 15 and 35 red, producing two subcases (the edge 15 blue and 25 and
35 red is not possible, as it would contradict the condition ii (red)).
Both situations are depicted in the figures above.
In the first subcase, by i (blue, 2), the edges 24 and 26 are blue. To make the blue copy isomorphic to I,
the remaining two blue edges must be 13 and 16. The red copy is then uniquely determined and isomorphic
to I, thus we get a decomposition of K6 into three copies of I.
In the second subcase, by i (blue, 3), the edges 13 and 36 are blue. To follow i (blue, 2) and not to
contradict ii (blue), the edge 26 must be blue. The blue copy and hence also the red copy are then uniquely
determined and we get a required decomposition of K6 .
Case 2.
In this case, by i (blue, 5), i (red, 5) and ii (blue) either the edge 25 is blue and the edges 15 and 35 are
red or the edge 15 is blue and the edges 25 and 35 are red, giving again two subcases.
In the first subcase, to satisfy ii (red), the edge 24 must be blue. Then the edge 26 is red. Again to satisfy
ii (red), the edge 46 must be blue and so the edge 14 red. Then the red copy, hence also the blue copy is
uniquely determined. We get a third decomposition of K6 into three copie of I.
In the second subcase, by i (blue, 2), the edges 24 and 26 are blue. To follow ii (blue), the edge 13 must
be red and so the edge 36 blue, determining the blue copy. The red copy turns out to be isomorphic to I as
well, thus we get another decomposition of K6 .
Case 3.
In this case, again by i (blue, 5), i (red, 5) and ii (red), either the edge 15 is red and the edges 25 and
35 are blue, or the edge 35 is red and the edges 15 and 25 are blue, giving the two subcases depicted in the
figures below.
21
In the first subcase, by i (red, 3), the edges 13 and 36 are red. Then the remaining two red edges must
be 46 and 24. The blue copy is then determined as well and turns out to be isomorphic to I. We get a fifth
decomposition of K6 into three copies of I.
The second subcase splits into three cases a, b and c , according to the choice of the blue edge incident
to the vertex 4. All three possibilities are depicted below.
In case a we picked the edge 24 to be blue and edges 14 and 46 to be red. Then by i (red, 2), the edge 26
is red. The red copy, hence also the blue copy, is then uniquely determined and isomorphic to I. We get a
decomposition of K6 into three copies isomorphic to I.
In case b we picked the edge 14 to be blue and edges 24 and 46 red. Then the blue copy is immediately uniquely determined. The red copy turns out to be isomorphic to I as well and so we get a seventh
decomposition of K6 .
Finally, in case c we picked the edge 46 to be blue and edges 14 and 24 to be red. The remaining two red
edges must then be the edges 16 and 36. The blue copy is then determined and isomorphic to I. We get the
eighth decomposition of K6 .
To sum up all the cases above, once one copy of the graph I is fixed, there are eight ways to finish the
decomposition. Therefore, n = 8.
Hence, there are 360·8
3 , i. e. 960 different systems. A reaction graph corresponding to the subgraph I has
960 vertices.
22
4
Reaction graphs corresponding to subgraphs A, B and C
In Section 3 we found the number of vertices in reaction graphs corresponding to all possible subgraphs
of K6 for which K6 decomposes into three isomorphic copies. The results are summarised in the following
table.
subgraph
no. of vertices in a
reaction graph
A
15
B
120
C
120
D
30
E
240
F
240
G
240
H
240
I
960
Reaction graphs corresponding to the subgraph D, also known as the graph K4 − e, have already been
investigated in []. Following this example, the aim of the rest of this project is to construct the reaction
graphs corresponding to subgraphs A, B and C and investigate some of their properties. The reaction graphs
corresponding to subgraphs E to I have all order ≥ 240 and are therefore left for further study.
Section 4.1 outlines the general methodology, discusses the issues common to the reaction graphs of all
three subgraphs A, B and C and identifies the properties which are studied in the next three Subsections.
4.1
Reaction graphs: methods of construction, definitions of adjacency and
investigated properties
In order to find the number of vertices in reaction graphs, we only needed calculations by hand. To
produce and manipulate the reaction graphs, however, we take the advantage of computational power. The
reaction graphs investigated in this project have been generated by the following method.
Starting with a given a subgraph (A, B or C), the program produces all the individual decompositions
of K6 (15 in case of A and 120 in cases of B and C). The subgraph is stored in the form of an edge list (i.
e. a listing of all edges in the subgraph) and so, by permuting the vertex labels, we get all possible labellings
of one copy of the subgraph on six vertices. Given the labellings, a backtracking algorithm is then used to
form the systems.
The algorithm works as follows: it orders the fixations into a list a and creates a list b of fixations from
a disjoint from the first fixation in a. In the very same way it subsequently creates a list c from the list b.
The list c is either empty, or the fixations placed first in all three lists form a system which we store. In
either case we then delete the first fixation from the list b (and also from a, if the list b gets empty after the
first deletion). The process with creating a list c (and possibly b as well) is then repeated several times until
all the entries in the list a have been used up, at which point we have found all the systems. A list of all
systems, the vertices of a reaction graph, is stored.
At this point we need the definition of adjacency in our reaction graphs. As explained in Section 1, edges
in the reaction graphs are determined by defining a ‘small change’. Two systems are adjacent in the reaction
graph if one can be obtained from another by carrying out the specified change. In this project we consider
the following two types of such a small change:
1. an overlap on significant edges, i. e. two decompositions of K6 are adjacent if they overlap on some
specified edges (possibly an empty set), and
2. a transposition, i. e. two decompositions of K6 are adjacent in the reaction graph if one can be obtained
from the other by a transposition of vertices.
We apply the above definitions to the three sets of vertices at hand, namely, the decompositions of K6
into copies of A, B and C. The resulting reaction graphs are then studied in detail in the following three
Subsections. Notice that these are only some examples of the reaction graphs corresponding to subgraphs A,
B and C: although the vertex sets are in each case firmly determined, the edges depend on the definition of
the adjacency.
By the same token, the definitions stated above produce for a given subgraph, in general, different
reaction graphs. Depending on the subgraph, however, the reaction graphs may be related or even identical.
23
Comparisms of definitions of adjacency for subgraphs A, B and C are one of the issues investigated in the
next Subsections.
Returning to the practical construction of the reaction graphs, we see that the procedures for generating
the reaction graphs are obviously dependent on the adjacency criterion chosen and the subgraph in question;
but, roughly speaking, they mostly involve a systematic search of pairs of systems for those which satisfy the
chosen adjacency criterion (and thus, form an edge). The reaction graphs are then stored in the form of an
adjacency matrix with ‘a list of vertices’, i. e. the corresponding systems.
We aim to describe some basic properties of our reaction graphs, such as the degree and connectivity, as
well as the ways of visualisation of the individual reaction graphs. Adjacency matrix and the computational
techniques make these investigations easy. The traditional combinatorial arguments are therefore occasionaly
supplemented by program computations and the results checked on computer.
4.1.1
Isomorphic systems, regular graphs
Two decompositions of K6 are isomorphic if there is a permutation of vertices which takes one decomposition into the other. We claim that in the case of subgraphs A, B and C, their corresponding sets of
systems consist entirely of pairwise isomorphic decompositions. If we prove this claim, it easily follows that
all the reaction graphs under investigation (i. e. those obtained from subgraphs A, B or C and adjacency
type 1 or 2) are regular. Regularity is an important property that makes many further calculations easier.
The exact degrees of individual reaction graphs are provided later.
Let’s prove the above claim. Given two systems corresponding to a subgraph S, there certainly exists a
vertex permutation that makes the systems coincide on at least one copy of S. Therefore, to prove the above,
it suffices to fix one copy of S and show that any two systems that contain the fixed copy are isomorphic.
This clearly holds for the subgraph A, since as we proved in Section 3.1 there always is only one decomposition containing a given fixed copy.
In case of the subgraph B, if we fix the black copy as in Section 3.2, the following are the only systems
containing it.
The first and the second system are isomorphic by permutation (12)(36)(45), the third and the second system by (56), and similarly the first and the forth system by permutation (56). Thus, all systems
corresponding to the subgraph B are pairwise isomorphic.
Finally, in case of the subgraph C, if the black copy is fixed as in Section 3.3, there are two systems
containing this copy, shown below.
.
These systems are isomorphic by a transposition (23). Thus, all the systems corresponding to the graph
C are also pairwise isomorphic.
24
Note that although many subgraphs, similarly as A, B and C, produce only isomorphic systems, this is
not always the case. For example, the following two decompositions corresponding to the subgraph I are not
isomorphic.
4.2
Reaction graph corresponding to subgraph A
Recall that in Section 3.1 we proved that the black, red and blue isolated edges of the graph A form
a perfect matching of K6 and also that the system is determined by fixing the black copy of A. Since the
matching uniquely determines the black copy (and vice versa), each system is, in fact, also determined by
specifying a matching. Thus, the vertices of the reaction graphs can be interpreted as the 15 ways in which
a matching can be chosen on six vertices.
To define the adjacency, we need to pick a “significant edge” of the subgraph A. A natural choice is to
take the isolated edge. There are three of these edges in each system, one from each copy of A. It is easy
to see (consider the matching) that any two different systems share either zero or one isolated edge. We
therefore define two systems in a reaction graph to be adjacent if they coincide on one of their isolated edges;
disjoint otherwise. Another possibility would be to define two systems as adjacent if one can be obtained from
another by a transposition. We claim that in case of the subgraph A these two definitions of adjacency are
equivalent. Indeed, if two systems share wlog the black isolated edge then their respective perfect matchings
must be some two of the following three:
And any two of these matchings are related by a transposition. Conversely, a transposition affects at
most two edges, therefore any two systems related by a transposition also share an isolated edge. We thus
consider here only the adjacency defined by the overlap on an isolated edge.
We already know that the reaction graph R1 corresponding to the subgraph A is regular. By generating
the adjacency matrix of the graph on a computer and couting the nonzero entries in each row of the matrix we
easily find that the degree in the reaction graph is six. This result is confirmed by a combinatorial argument:
given a matching M , there are two other matchings that coincide with M on the black edge. Since a matching
can coincide with M on any of its three edges, the system has 6 neighbours in total.
Next we find the number of neighbours that two adjacent systems have in common. A computer computation shows that this number is equal to one for any pair of adjacent systems. Indeed, if two matchings
M1 and M2 coincide on wlog the black edge, then how many matchings M3 are adjacent to both M1 and
M2 ? If M3 shared with M1 an edge e other than the black one, then since M1 and M2 are different systems
that share the black edge, e must in M2 connect the red and the blue edge. Then M2 and M3 can only
coincide on the black edge. But that implies that also M1 and M3 coincide on the black edge, meaning that
M1 = M3 , contradiction. Therefore M3 shares with both M1 and M2 the black edge. There exists only one
such matching different from both M1 and M2 , hence M1 and M2 only have one neighbour in common.
25
Similarly we calculate the number of systems common to any two non-adjacent matchings. Suppose that
the matchings M1 and M2 do not share any isolated edge. Choosing an edge e of M1 that M1 and M3 share
uniquely determines the matching M3 . This is because e, similarly as above, connects in M2 two of its edges
and only the third one may then be shared with M3 . Since there are three ways to choose e in M1 , there
exist three systems adjacent to both M1 and M2 .
We have thus proved that the reaction graph R1 is the strongly regular graph srg(15, 6, 1, 3), cf. [1].
Its complement, the strongly regular graph srg(15, 8, 4, 4) corresponds to the case when two systems in the
reaction graph are defined to be adjacent whenever their perfect matchings have no edge in common.
Using the adjacency matrix, we generated a drawing of R1 :
Another drawing of R1 , including also the matchings corresponding to the individual systems, has been
presented in [Wolfram]:
4.3
Reaction graphs corresponding to subgraph B
Recall from Section 3.2 that in each system corresponding to subgraph B, the black, red and blue main
edges form a perfect matching. There are 120 systems in total and 15 ways in which a matching can be
chosen on six vertices. Therefore, each matching M must occur in eight systems. Call these systems the
octuple corresponding to a matching M .
We now consider two types of adjacency, creating two different reaction graphs R2 and R3 corresponding
to the subgraph B.
26
4.3.1
Adjacency defined by an overlap
To define the first type of adjacency, we need to pick a significant edge of the subgraph B. A natural
choice is to take the main edge. There are three main edges in each system and since they form a perfect
matching, any two different systems overlap on either 0, 1 or 3 main edges. In the graph R2 define two
systems to be adjacent if they overlap on at least one main edge. We aim to understand the structrure of
R2 .
Notice that R2 decomposes in two subgraphs S and T , both with the same vertex set as R2 , and such
that two systems are adjacent in S if they share exactly one main edge and in T if they share exactly three
main edges. By describing the reaction graphs S and T we can understand R2 .
A system in T is adjacent exactly to systems with the same perfect matching. Since there are always
eight systems having the same matching, T is regular of degree seven and each octuple corresponding to a
given matching forms a complete graph K8 . Thus, T is a disconnected graph consisting of 15 copies of K8 .
Unlike T , S contains only edges between different octuples. Also, since the adjacency in S only depends
on the perfect matching and ignores all other edges in the system, whenever a system from octuple O1 is
adjacent to a system in octuple O2 , then, in fact, every system in O1 is adjacent to every system in O2 . Call
such octuples adjacent. Each pair of adjacent octuples then forms a complete bipartite graph K8,8 .
Finally, to see which octuples are adjacent and which not, we need to find which pairs from the 15 different
matchings corresponding to individual octuples overlap on one main edge. But this is exactly the problem
solved in Section 4.2 for the subgraph A with the resulting reaction graph R1 ; therefore, two octuples are
adjacent if and only if their corresponding matchings are adjacent in the graph R1 . Since R1 is regular of
degree six, each octuple is adjacent to six other octuples, i. e. 48 different vertices. The graph S is regular
of degree 48. The fact that both R1 and K8,8 are connected implies that also S is connected.
We can now describe the reaction graph R2 . R2 turns out to be a blow-up of the strongly regular graph
R1 obtained by replacing each vertex by a copy of K8 and each edge by K8,8 . For any choice of 15 systems,
one from each octuple, the systems form the graph R1 . A drawing of R2 is suggested in the following figure:
the 15 octuples are joined by the ‘edges’ to create the graph R1 , where each ‘edge’ is, in fact, a complete
bipartite graph K8,8 betwen two copies of K8 .
The reaction graph R2 is regular of degree 55 (= 7 + 48) and connected (since S is). Unlike R1 , however,
the reaction graph R2 is not strongly regular.
27
4.3.2
Adjacency defined by a transposition
Another possibility is to define the adjacency by a transposition. In the reaction graph R3 let two
systems be adjacent if one can be obtained from the other by a transposition of vertices.
Since a transposition affects at most two of the three main edges forming a perfect matching in a system,
any two systems which are related by a transposition must also overlap on at least one of their main edges.
Therefore, the graph R3 is a subgraph of R2 .
Given a system S, wlog assume that its main edges are 12, 34 and 56, there are always three transpositions
- in this case (12), (34) and (56) - that fix all the main edges, thus producing in the reaction graph R3 edges
within one specific octuple of systems. Moreover, we claim that the vertex permutation exchanging the
endvertices of the main edges, i. e. in this case the permutation (12)(34)(56), fixes all the systems in the
given octuple.
Indeed, notice that in each system, taking a copy of the subgraph B and the remaining two main edges,
these form two 4-cycles in which every second edge is a main edge. Permuting the endvertices of each main
edge then fixes both 4-cycles, hence also the copy. This holds for all copies of B in the system, therefore the
system, as well as all other systems with the same set of main edges, stays fixed.
Therefore, applying a permutation (12)(34) to the system S is the same as applying the transposition
(56), similarly, (12)(56) is the same as (34) and (34)(56) as (12). Following this argument for other systems
in the same octuple (whose neighbours are always obtained by transpositions (12), (34) and (56)), we see
that the systems within the octuple form two disjoint copies of a K4 . Since all the systems corresponding to
the subgraph B are isomorphic, any other octuple in the reaction graph R3 consists of two copies of K4 as
well.
We now discuss the edges between two different octuples O1 and O2 of R3 , whose respective sets of the
main edges overlap on one edge. Assume wlog that the main edges in O1 are 12, 34 and 56, while in O2 they
are 12, 36 and 45. Then there are two transpositions which transform the perfect matchings between O1 and
O2 , namely (35) and (46). Starting from a system S1 in O1 and carrying out in row the transpositions (35),
(46), (35) and (46), we get the systems S2 , S3 and S4 in the octuple O2 , O1 and O2 respectively, and finally
finish back at S1 . The systems S1 to S4 are clearly all different and so form a 4-cycle. Also, using a computer
program we see that the systems S1 and S3 (and similarly S2 and S4 ) are not related by a transposition,
28
and belong to different copies of K4 in O1 (or in O2 ). Applying these arguments to the remaining systems in
O1 and O2 and generalising, we see that any pair of ‘adjacent’ octuples in R3 is connected by four disjoint
4-cycles. Finally, notice also that any octuples that were adjacent in R2 are adjacent in R3 as well.
Thus, summing up all the above, the reaction graph R3 is again a blow-up of R1 , but in this case each
vertex of R1 is replaced by two copies of K4 and each edge is replaced by four 4-cycles. A drawing of R3 is
suggested in the following figure.
Each system in R3 is adjacent to 3 other systems in the same octuple and to two systems in each of six
other octuples (since R1 has degree six). Thus, R3 is regular of degree 15. Similarly as R2 , neither R3 is
strongly regular. Since R1 is connected and any two adjacent octuples also form a connected graph, R3 is
connected as well. Finally, as has already been mentioned above, R3 is a subgraph of R2 . The reason for
R3 being much less dense than R2 is that while adjacency defined by an overlap ignores all edges except the
three isolated edges in a system, a transposition takes into account all edges of the system.
4.4
Reaction graphs corresponding to subgraph C
By looking at the two systems that we found in Section 3.3 for a fixed black copy, and by using the
fact that any two systems corresponding to the subgraph C are isomorphic, we see that the black, red and
blue isolated edges form in each system corresponding to subgraph C a 3-cycle. Call this 3-cycle the isolated
triangle. Its vertices are exactly the three main vertices of the system (see Section 3.3 for definitions of the
main vertex / isolated edge in subgraph C). Call the 3-cycle spanned by the remaining three vertices of the
system the free triangle. It turns out that by fixing the isolated triangle, the rest of the system is determined
except for the colouring of the free triangle. The free triangle has to have a black, red and blue edge, but
otherwise any of the six colourings satisfying this criterion makes a system. Hence, there are always six
different systems having the same isolated triangle. The 120 vertices of the reaction graphs split according
to their isolated triangles into 20 sextets.
29
We now consider two types of adjacency, creating two different reaction graphs R4 and R5 corresponding
to the subgraph C.
4.4.1
Adjacency defined by an overlap
Similarly as before, to define the first type of adjacency, we need to pick a significant edge in the subgraph
C. A natural choice is to take the isolated edge. Since the three isolated edges form a triangle in each system,
two systems may overlap on 0, 1 or 3 isolated edges. Define two systems in R4 to be adjacent, if they overlap
on at least one isolated edge.
To understand the structure of R4 , it is again helpful to think about the systems overlapping on three
isolated edges and those overlapping on one isolated edge separately. Overlap on all three isolated edges
happens exactly between any pair of systems in the same sextet, producing thus 20 disjoint copies of a K6 .
As for the overlap on exactly one edge, notice that whenever two systems from two different sextets are
adjacent, then, in fact, any two systems, one from each of these sextets, are adjacent. Neighbouring sextets
are therefore connected by a complete bipartite graph K6,6 .
It remains to determine which sextets are adjacent. For this, we consider a graph S on 20 vertices, in
which the vertices represent all possible subsets of order three of a 6-element set and in which two vertices
are adjacent if the corresponding subsets overlap on exactly two members. Since the isolated triangle characterising each sextet in R4 is determined by the choice of the three main vertices in the system, we see that
there is a bijection between the sextets of R4 and the vertices of S such that two sextets are adjacent in R4
if and only if the corresponding vertices in S are adjacent.
A drawing of S is in the following figure:
30
Given a 3-element subset T of a 6-element set U , there are 3 · 3 other subsets of U that overlap with T
on exactly two members, hence, the graph S is regular of degree nine. Also, S is connected; but unlike the
graph R1 , S is not strongly regular.
Putting all the above arguments together, we see that the reaction graph R4 is a blow-up of the graph S
in which every vertex is replaced by a copy of a K6 and each edge by a K6,6 . R4 is regular of degree 59 (each
system is adjacent to 5 systems within the same sextet and to six systems in each of nine different sextets),
connected, but not strongly regular. A drawing of R4 is suggested in the following figure.
4.4.2
Adjacency defined by a transposition
In the reaction graph R5 , define two systems corresponding to the subgraph C to be adjacent if one can
be obtained from the other by a transposition of vertices. By the same argument as used in Section 4.3 for
graphs R3 and R2 , also R5 must be a subgraph of the graph R4 . We aim to understand the structure of R5 .
We split the transpositions acting on a given system into three kinds:
Case 1.
transposition of two vertices in the free triangle of the system
Case 2.
transposition of two vertices in the isolated triangle of the system
Case 3.
transposition of a vertex from the free triangle with a vertex from the isolated triangle
We now discuss what kind of edges the above transpositions produce.
Case 1.
In this case the isolated triangle stays fixed and so the colourings of all edges except those in the free
triangle remain unchanged (remember that the vertices of the isolated triangle are the main vertices in each
system). In the free triangle, one of the edges keeps its colour, while the colourings of the remaining two
edges are swapped.
Thus, this kind of a transposition produces in R5 only edges within the individual sextets. In a given
sextet, each system T is adjacent to three other systems (corresponding to the three possible transpositions
of vertices of the free triangle), these are exactly the systems with which T shares colouring of one edge of
the free triangle.
Edges obtained by a transposition of type (i) in a sextet of R5 are shown in the following figure. Call the
resulting graph H. The vertices in the figure show the colouring of the free triangle in the respective systems.
31
Case 2.
Transposition of vertices of the isolated triangle fixes the position of the isolated triangle but swaps the
colourings of two of its edges as well as the colourings of the two triplets of unicoloured edges incident to the
swapped main vertices. The colouring of the free triangle remains unchanged. Since the actual colours of
the individual copies in a system do not matter, this is the same as fixing all edges in the system and only
swapping colours of two edges in the free triangle.
Thus, by the type (ii) transposition we only get edges that we already obtained in the previous case.
Case 3.
Transposition of a vertex from the isolated triangle with a vertex from the free triangle changes the
position of the isolated triagle and so in R5 it produces edges between systems from different sextets. We
claim that any two sextets S1 and S2 whose corresponding isolated triangles overlap on one edge (i. e. in R4
they are connected by a K6,6 ), are in R5 connected by a perfect matching.
Indeed, there is only one transposition that maps the vertices of the isolated triangle of S1 to the isolated
triangle of S2 . Thus given a system in S1 , its only neighbour in S2 is uniquely determined. Also, two systems
in S1 cannot be adjacent to the same system in S2 since that would mean that from one system in S2 we
may get two different systems in S1 by carrying out the same transposition, contradiction.
Finally, notice also that this transposition transforms adjacent systems in S1 into systems adjacent in S2 .
Thus, all in all we see that the reaction graph R5 is again a blow-up of the graph S. In this case each
vertex is replaced by a copy of the graph H and each edge by a perfect matching. A drawing of R5 is
suggested in the following figure.
32
In R5 , each system is adjacent to three other systems in the same sextet and to one system from nine
different sextets (since the graph S has degree nine), hence, R5 is regular of degree 12. As R4 , neither R5 is
strongly regular. Since the graph S is connected and any two adjacent sextets also form a connected graph,
R5 is connected as well. Finally, as in the Section 4.3.2, also here the reason for R5 being much less dense
than R4 is that while adjacency defined by an overlap ignores all edges except the three isolated edges in a
system, a transposition takes into account all edges of the system.
33
5
Conclusions and further study
In this project we have studied reaction graphs for some decompositions of a K6 . Our work was inspired
by an earlier chemical problem and followed on a paper by Lindner and Rosa [5]. We first managed to find all
the graphs on six vertices for which K6 decomposes into three isomorphic copies. For each of these graphs we
subsequently calculated the number of different decompositions of K6 , finding thus the number of vertices in
the respective reaction graphs. The reaction graphs corresponding to three of the subgraphs, namely those
producing the smallest orders of the reaction graphs and not investigated previously, denoted A, B and C,
were chosen for further study.
We defined the adjacency in our reaction graphs in two different ways: either by a transposition of vertices
or by an overlap on at least one significant edge, where, depending on the subgraph, the significant edge was
chosen to be either the isolated or the main edge of the subgraph.
In case of the subgraph A, the two definitions of adjacency coincided and so we only obtained one reaction
graph R1 , which turned out to be the strongly regular graph srg(15, 6, 1, 3).
In cases of B and C we obtained two pairs of reaction graphs, corresponding to two types of adjacency
and denoted R2 to R5 , each having 120 vertices. Reaction graphs R2 and R3 , corresponding to the subgraph
B, were both blow-ups of the graph R1 . The former was obtained by replacing each vertex of R2 by a copy
of K8 and each edge by a K8,8 . In the latter each vertex of R1 was replaced by two disjoint copies of K4 and
each edge by four disjoint 4-cycles.
Reaction graphs R4 and R5 , corresponding to the subgraph C, turned out to be blow-ups of a graph S.
S is a 9-regular, but not strongly regular, connected graph on 20 vertices, whose vertices may be thought of
as all the 3-element subsets of a 6-element set and edges as overlaps between two such subsets on exactly two
elements. Reaction graph R4 was obtained by replacing each vertex of S by a copy of K6 and each edge of
S by a K6,6 . R5 was obtained by replacing each vertex of S by a graph H on eight vertices and each edge of
S by a perfect matching.
All the reaction graphs R2 , R3 , R4 and R5 were found to be regular, of degrees 55, 15, 59 and 12
respectively. Each of them is connected, but none is strongly regular.
So far we investigated the basic structure of the reaction graphs R1 to R5 . There are many more properties,
which could be looked at in further research; for example, the automorphism group of the reaction graphs
and combinatorial properties of these graphs, such as the diameter, girth, independence number, clique
number, hamiltonicity and so on. Attention in future could also be paid to the reaction graphs of higher
orders corresponding to the other subgraphs for which we found that K6 decomposes into three isomorphic
copies. Finally, K6 can also be decomposed into five isomorphic subgraphs, each having three edges, and thus
producing new reaction graphs. Reaction graphs for decompositions of complete graphs of orders different
from six may also be studied, of course.
34
References
[1] A. E. Brouwer, Strongly regular graphs in CRC Handbook of Combinatorial Designs, 2nd edition (Eds.
C. J. Colbourn, J. H. Dinitz), VII.11, CRC Press, Boca Raton, 2007, pp.852-868.
[2] F. Harary, Graph theory, Addison-Wesley, 1969, p.219.
[3] M. H. Klin, S. S. Tratch, N. S. Zefirov, Group-theoretical approach to the investigation of reaction graphs
for highly degenerate rearrangements of chemical compounds. I. Criterion of the connectivity of a graph,
Journal of Mathematical Chemistry, 7 (1991), 135-151.
[4] M. H. Klin, N. S. Zefirov, Group theoretical approach to the investigation of reaction graphs for highly
degenerate rearrangements of chemical compounds. II. Fundamental concepts, MATCH, 26 (1991), 171190.
[5] C. C. Lindner, A. Rosa, Reaction graphs of the K4 − e design of order 6, Bulletin of the ICA, 47 (2006),
43-47.
[6] E. K. Lloyd, The reaction graph of the Fano plane, Combinatorics and Graph Theory ’95, Vol 1, pp.260274
[7] E. K. Lloyd, G. A. Jones, Reaction graphs, Acta Applicandae Mathematicae, 52 (1998), 121-147.
[8] http://mathworld.wolfram.com/GeneralizedQuadrangle.html, accessed on 1/6/2012
35