On Unitary Cayley Graphs
Italo J. Dejter
University of Puerbo Rico
Department of Mathematics
Rio Piedras PR 00931
Reinaldo E. Giudici
Universidad Sim6n Bolivar
Departamento de Matem6ticas
Caracas, Venezuela
AgsrRAcr. We deal with
a family of undirected Cayley graphs
X* which are unions of disjoint Hamilton
cycles, and some of
It is
their properties, where n runs over the positive integers.
proved that X* is a bipartite graph when n is even. If n is a"n
odd number, we count the number of different colored triangles
in X..
1. Introduction
The graphs considered in this paper are undirected, simple and without
loops. Given a graph X"r, we denote its vertex set and edge set by V(X)
and E(X), respectively. Given a positive integer ro and an element r of the
additive cyclic group zn of integer residue classes mod 2, there is only one
integer representative y of c such that 0 < y < n; we use the same symbol
c to denote gr and define lrl :, if a < nl2 and lcl : rt - a if t > nf2The group Z* possesses the subset Un : U(Zn) of units modulo n,
constituted by those t in Zn with integer representatives relatively prime
to n. It is known that [/, is a multiplicative group ' Let Wn: UnlZzbe
given by the classes {*, -r}, where z varies in U'. Notice thatWn inherites
a group structure ftom (Jn. We represent each {r, -t} € Wn again by t,
where 0 < r ( [n/2], unless confusion arises.
We deal with a class {X'} of undirected Cayley graphs X', that we call
"Unitary Ctyley graphs", defined as follows: V(X*) : Zn and any two
vertices ?.r1 and a2 are adjacent if and only if lr, - ozl e [/'. Each edge
JCMCC
l8
(1995), pp.
t2t-t24
of Xn is attributed a color from the ,"t
{J, 2, ...,k}, where k : [n/Z]
according to the following rule: If e is an edge with
e'nivertices u1 a,'d. ai
and if o1
*i (mod ?), i e {!,2,... , lc},-then ."
it" edge e witt
Laz:
i' Accordingtothe definition of xn, we have that i €"otor
[/,n. Therefore, x,
i1tng_ylairected Cayley graph of Zn with generator v':t, Wn. Written Caj
l!",w^l,hence, we may say that x, has an edge coloring ,uith o.ru hatf oj
the unit element of Zn. For intance, if n. : g th; U* :
g} and
Wn: {1,2,4}. Thus, Xs: Cay [Zg,Wsl is represented {'7,2,;:;,2,
in the Figure 1.1.
In [1] Dejter dealt with Cayley graphs of the form Cay
where
1, is the generator s9J {1,2,...,i}, n is odd and ft : [Zn,In],
jlt2,
i.e.,
6,':
the complete graphs 1(* edge-colo."d i., a symmetric fashion,
and studied
some induced subgraphs K" of these cayley graphs
that were called totally
multicolored (TMC) subgraphs, motivated by a questio'
of E.dO", pyber
and T\rza [2].
65
Figure 1.1
In the present paper, we deal with some properties of the X,".
In partic_
ular, besides proving that if , is even, then x, is bipartite
wL count the
number of different colored triangles of Xn, when n is
odd.
2. Basic Properties of Unitary Cayley Graphs
we now state some properties of the unitary cayley graphs.
The
are easy and left to the reader.
X, is a regular graph
Euler g-tunction, and is a union of
66)
any positive integer n.
Proposition 2.L.
of
degree
d(n),
u"iitt"n'iyiin
proofs
g(n) is the
or teigtn n, for
where
Proposition 2.2. Xn is isomorphic to a complete graph with n ver_
ticest il n is,a prime number and. isomorphic to a comprete
bipafiite graph
K(2t-t,2t-r) if n-2t.
122
The graphs X* for n' :2t are interesting from the chromatic point of
view. They are regular of degree one half of the number of vertices and the
number of edges turns out to be the square ofthe degree ofthe graph. These
graphs are a special type studied in [3]; they are chromatically unique.
Unitary Cayley graphs have different properties according to the parity
of n, as will be seen in the next proposition.
Proposition 2.3. Xn is a bipartite
graph
if n is
ant
even number.
of Znrnust be odd. Thus, no two even labeled
vertices are adjacent. This implies that the even labeled vertices and the
tr
odd labeled vertices form a bipartition of the vertex set.
Proof:
Since n is even, units
Corollary 2.3.1. If n is even, then Xn
particular, Xn is triangle-free.
has no odd length cycles. In
3. Number of tiangles in X.
since X* is triangle-free for n even, we will
consider ru to be an odd number
in this section. Let us denote by {o, b,c} a triangle in X' with vertices a,
b and c;therefore t(b-a), *(c-b), +(a-c) are elements inUn' Without
loss of generality we may assume that our triangles have vertices {0, 1,u},
u € (Jn.If we denote by ?or the set of all the triangles having the common
vertices 0 and 1 i.e., Tor : {{0, 1,r} I u e Un}, then the cardinality
procedures it
lTorl : {u e Un I (" - t) € U^}. Using elementary counting
can be proved that
l?orl:
"y^(t -:)
To count the number of triangles in x, let us consider the action of
the group G : (In x Zn on the set of triangles of X' i.e., if. (u,a) e G,
then (u,z){0,1, u}: {ur,o(1+r), u(u*r)}' Each orbit of the triangles
corresponding to the pair (u, r) may have at most six different elements
(0,1,;), (0,tl--r,1), (1 -r)-1,0,1), (1,(r^,,- 1)u-1,0), (u(u- 1)-t,1,0) in
Tor. It can happen that some orbits have exactly 3,2 or even 1 elements.
We have l"orl : Dotedha, where h6 is the number of orbits with exactly
d elements in Tor. fhe orbits with d difigr.ent elements have a number of
: d(nli(nt. Thus, if ? stands for the
triangles given by lohdl : W
number of triangles of X', then
": D
haloal:
h6nQ(n) +
h"?9P + hrryP *
nrgP,
d/6
r:
n4@) (ne +
+.+.+) :d@)Y:nd@)*,
r23
where dz(n) : nDp/n (t - *) by its resemblance to the Euler ffunction.
Note that $2(n) is the number of consecutive units modulo n.
Finally, if we denote by IT the number of triangles in X, that have two
sides painted with equal colors, let us call these "isosceles-color-tria.ngles,'
and denote by ET the corresponding number of triangles with their three
sides painted in different colors, called totally multicolored triangles. With
obvious meaning of "scalene-color-triangles" we obtain the following result.
Proposition 3.1. If X^
is a Unitary Cayley graph with
i
t
')
n an odd number
then
rc:?gp
ad
nr:n6!n)
(dz(n) -s).
Proof:
Since n, is odd {0, 1, -1} is a isosceles color triangle; its orbit under
the group action gives all the others. It can easily be verified that d:3.
Thus the formula for IT follows.
For the scalene-color-triangles let us observe that there is no "equilateralcolor-triangles", i.e. triangles with its three sides painted with the same
color. Therefore, ET : T - IT.
D
Remark: Xg has 27 triangles and all of them
are isosceles-color-triangles.
The first graph having scalene-color-triangles for n different from a prime
number is Xzt. It has 126 isosceles-color-triangles, 84 scalene-color-triangles
and a total of270 triangles.
The authors are grateful for the suggestions ofthe referee which shortened
and improved the paper and for the help given by Professor P. Berrizbeitia
and the corrections of English given by our colleague Professor T. Berry.
Thanks to all of them.
References
[1] I.J. Dejter, Totally Multicolored, Subgraphs of Complete Cayley Graphs,
Congressus Num. 7O (1990). pp. 53-64.
[2] P. Erd<;s, Problems and Results ,i,n Comb,inatori,al Analys'i,s and Combinatori,al Number Theory, in "Graph Theory, Combinatorics and Applications", (ed. Y. Alavi), J. Wiley 1991, vol. 1 pp. 397-406.
[3] R.E. Giudici and E. Lima de 56, Chromati,c (Jniqueness of Certain
partite Graphs, Congressus Numerantium, 76 (1990), pp. 69-75.
r24
Bi-
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