c 1998 Society for Industrial and Applied Mathematics
SIAM J. DISCRETE MATH.
Vol. 11, No. 1, pp. 128–134, February 1998
009
COMPETITION GRAPHS OF HAMILTONIAN DIGRAPHS∗
DAVID R. GUICHARD†
Abstract. K. F. Fraughnaugh et al. proved that a graph G is the competition graph of a
hamiltonian digraph possibly having loops if and only if G has an edge clique cover C = {C1 , . . . , Cn }
that has a system of distinct representatives. [SIAM J. Discrete Math., 8 (1995), pp. 179–185]. We
settle a question left open by their work, by showing that the words “possibly having loops” may be
removed.
Key words. competition graph, hamiltonian digraph
AMS subject classification. 05C15
PII. S089548019629735X
1. Introduction. Suppose that D is a digraph. The competition graph or
conflict graph C(D) has the same vertex set as D, and an edge {u, v}, u 6= v, if
there is a vertex w such that (u, w) and (v, w) are arcs in D. Competition graphs
are useful in the study of such diverse systems as food webs and radio networks
and have received substantial attention over the past fifteen years. Characterizations
of competition graphs for a variety of classes of digraphs have been reported over
the years; recently, K. F. Fraughnaugh et al. [2] have given characterizations for
competition graphs of strongly connected digraphs and hamiltonian digraphs. Here
we improve their characterizations of competition graphs of hamiltonian digraphs.
(See the same paper for references to other characterizations.)
2. Hamiltonian digraphs. All digraphs are loopless and contain no multiple
edges unless otherwise indicated. We use circuit to mean a directed cycle in a digraph.
Fraughnaugh [2] contains the following two characterizations.
Theorem 1. A graph G on n vertices is the competition graph of a hamiltonian
digraph if and only if G has an edge clique cover {C1 , . . . , Cn }, with a system of
/ Ci−1 . (Subscript arithmetic “wraps
distinct representatives {v1 , . . . , vn } such that vi ∈
around,” i.e., C0 = Cn .)
Theorem 2. A graph G on n vertices is the competition graph of a hamiltonian
digraph, possibly with loops, if and only if G has an edge clique cover {C1 , . . . , Cn }
with a system of distinct representatives.
We show that these characterizations can be “combined” as follows.
Theorem 3. A graph G on n ≥ 3 vertices is the competition graph of a hamiltonian digraph if and only if G has an edge clique cover {C1 , . . . , Cn } with a system of
distinct representatives.
Proof. One direction follows immediately from Theorem 1. The other direction
follows if we can show that whenever G has an edge clique cover {C1 , . . . , Cn } with
a system of distinct representatives, it has one satisfying the additional property of
/ Ci−1 . We show precisely this in Theorem 6. Note that
Theorem 1, namely, that vi ∈
∗ Received by the editors January 17, 1996; accepted for publication (in revised form) November
22, 1996.
http://www.siam.org/journals/sidma/11-1/29735.html
† Department of Mathematics and Computer Science, Whitman College, Walla Walla, WA 99362
([email protected]).
128
COMPETITION GRAPHS OF HAMILTONIAN DIGRAPHS
129
K2 has an edge clique cover with a system of distinct representatives but is not the
competition graph of a hamiltonian digraph.
Suppose that G is a graph on n vertices and C = {C1 , . . . , Cn } is a clique cover
of G with a system of distinct representatives V = {v1 , v2 , . . . , vn }. The clique graph
CG(C, V) is the digraph whose vertices are the cliques, with arc (Ci , Cj ) if and only if
vj ∈
/ Ci . To complete the proof of Theorem 3, we want to show that if G has a clique
cover with a system of distinct representatives, then it has one whose clique graph is
hamiltonian. (For if the clique graph is hamiltonian, we may renumber the cliques, if
necessary, so that (C1 , C2 , . . . , Cn ) is a circuit. By definition of the clique graph, this
implies the “extra condition” of Theorem 1.)
Recall the well-known theorem of Ghouila-Houri.
Theorem 4. Suppose G is a strongly connected digraph without loops or multiple
+
−
arcs. If d(v) = d (v) + d (v) ≥ n for every vertex v, then G is hamiltonian.
We will use this in two ways: in some special cases, we will be able to invoke
Theorem 4 directly. For the rest, when the hypotheses of the theorem are not quite
met, we give a proof that is much like the proof of Theorem 4, using some properties
of the clique graph to make up for the failed hypotheses.
We will need the following technical lemma (also used in the proof of Theorem
4).
Lemma 5. Consider a circuit with m vertices, each colored either red or blue.
Suppose the circuit contains exactly nr ≥ 1 red vertices and nb ≥ 1 sequences of q
consecutive blue vertices. Then nr + nb ≤ m − q + 1.
This lemma, and the proof of Theorem 4 that we use, are from Berge [1].
DEFINITION. If C is a P
graph, let |C| denote the number of vertices in C. The
size of a set of cliques C is C∈C |C|.
Remark. When C is a set, not a graph, we use |C| in the usual sense to mean the
number of elements in the set C.
DEFINITION. If A = (VA , EA ) and B = (VB , EB ) are subgraphs of G, by A ∪ B
we mean the subgraph with vertex set VA ∪ VB and edge set EA ∪ EB .
Theorem 6. If G is a graph on n ≥ 3 vertices and has an edge clique cover
{C1 , . . . , Cn } with a system of distinct representatives, then it has one whose clique
graph is hamiltonian.
Proof. If G is a complete graph, the theorem is easy. For other G, we prove by
induction on the number of vertices that if G has a clique cover with a system of
distinct representatives, then among all such clique covers there is one of minimum
size whose clique graph is hamiltonian. The theorem is easy to prove for n = 3, so
suppose n ≥ 4.
If C is a clique cover with a system of distinct representatives, let k(C) ≥ 0 be the
largest integer
S strictly less than n such that some collection A of k(C) ofSthe cliques
has |A| = | A|. (Recall that Hall’s marriage
principle
S
S says that |A| ≤ | A| for all
A ⊆ C. Here and subsequently, we use A to mean C∈A C.)
Let C = {C1 , . . . , Cn } be a clique cover of G of minimum size with a system of
distinct representatives V = {v1 , v2 , . . . , vn }, for which k = k(C) is as large as possible.
Sk
We may assume that k = | i=1 Ci |.
Unless k = n − 1 and Cn is a singleton, the clique cover has the following “minimality property,” henceforth (MP): if v ∈ Ci , i > k, then there is an edge {v, w} in
Ci that is in no other clique, for if not, then v could be removed from Ci to form a
clique cover with a system of distinct representatives of smaller size. (Note that Ci
cannot be a singleton, by definition of k.)
130
DAVID R. GUICHARD
Remark. If every edge {v, w} in Ci is in some other clique, then it is clear that
removing v from Ci leaves a clique cover. It is perhaps not obvious that this new
cover has a system of distinct representatives. (Actually, it is clear if k = n − 1,
so we may assume that k < n − 1.) For a contradiction, suppose it doesn’t;
S by
Hall’s marriage principle, Ci must be in some set of cliques A such that |A| = | A|,
Sk
and Ci is the only clique in A that contains v. If v ∈ j=1 Cj , then v 6= vi , so
A with Ci replaced by Ci \{v} has a system of distinct representatives (inherited
Sk
from V), contradicting the definition of A. Hence, v ∈
/ j=1 Cj . Now there are two
cases: S
(1) If A ⊇ {Ck+1 , . . . , Cn }, then B = {C1 , . . . , Ci−1 , Ci+1 , . . . , Cn } satisfies
|B| = | B|, contradicting
the definition of k. (2) Otherwise, B = A ∪ {C
S1 , . . . , Ck }
S
satisfies |B| = | B|, contradicting the definition of k. To see that |B| = | B|, pick a
system of distinct representatives for A∪{C1 , . . . , Ck }. If there is a v ∈ Cj , j ≤ k that
Sk
is not one of the representatives, then k < | j=1 Cj |, contradicting the definitionSof k.
If there is a v ∈ Cj , Cj ∈ A that is not one of the representatives, then |A| < | A|,
contradicting the definition of A.
Here is an outline of the rest of the proof.
I. Establish some properties of C1 , . . . , Ck and of arcs between these cliques
and Ck+1 , . . . , Cn .
II. Show that d(C) ≥ n for C ∈ {Ck+1 , . . . , Cn }.
III. Show that CG(C, V) is strongly connected.
IV. Use Theorem 4 to show that CG(C, V) is hamiltonian in some special cases.
V. Show that CG(C, V) is hamiltonian using methods similar to the proof of
Theorem 4.
I. Some properties of C1 , . . . , Ck .
We establish some properties of C1 , . . . , Ck , in some cases by replacing C by a
different clique cover.
If k > 0, the cliques C1 , . . . , Ck form a clique cover with a system of distinct
representatives for a smaller graph G′ , namely, the union of the cliques C1 , . . . , Ck .
If G′ is not a complete graph, then, by the induction hypothesis, we may replace
C1 , . . . , Ck by a clique cover of G′ , with a system of distinct representatives, that has
the same size as C1 , . . . , Ck , and whose clique graph is hamiltonian. For convenience,
call the new cliques C1 , . . . , Ck as well.
If not all of C1 , . . . , Ck are singletons, every clique Cg , g > k must have an arc
(in CG(C, V)) to some Ci , i ≤ k. If not, Cg contains all of v1 , . . . , vk , so we could
replace the cliques C1 , . . . , Ck by singletons to get a smaller clique cover, which is a
contradiction.
If G′ is a complete graph and k ≥ 2, the cliques C1 , . . . , Ck must consist of k − 1
singletons and a copy of Kk , because C was chosen to have minimum possible size.
To see this, suppose that no Ci , i ≤ k is Kk . Then each v in G′ must be in at least
two of the Ci for i ≤ k, and so the size of the cover C1 , . . . , Ck is at least 2k, while
the size of k − 1 singletons and of a Kk is 2k − 1. We may assume that C1 , . . . , Ck−1
are the singletons. We also may assume that some Cg , g > k does not contain C1 , by
the previous paragraph, and by some renumbering of the vertices of G′ if necessary.
To see this, note that if every Cg , g > k contains all of C1 , C2 , . . . , Ck−1 , then no Cg ,
g > k contains vk , the representative of Ck . Replacing C1 by {vk } and renumbering,
we get a set of cliques with the desired property. Finally, we may assume that some
Cg , g > k, has an arc to Ck ∼
= Kk . For if not, then every Cg , g > k contains all
of v2 , . . . , vk and does not contain v1 . Then, if k > 2, we may replace C1 , . . . , Ck
COMPETITION GRAPHS OF HAMILTONIAN DIGRAPHS
131
by {v1 , v2 }, {v1 , v3 }, . . . , {v1 , vk }, {v2 }. Together with Ck+1 , . . . , Cn , these form a
clique cover with a system of distinct representatives of the same size as the original,
so we may use this clique cover instead of the original. For convenience, name this
new clique cover C, and name the new cliques C1 , C2 , . . . , Cn . If k = 2, then we can
′
replace C1 = {v1 } and C2 = {v1 , v2 } by C1′ = {vS
2 } and C2 = {v1 , v2 }, and then
′
A = {C1 , C3 , . . . , Cn } has the property that |A| = | A|, which is a contradiction, by
the definition of k.
If C1 , . . . , Ck are all singletons and every Cg , g > k contains all vertices of G′ , let
v be a vertex of G that is not in all of the cliques Cg , g > k. (If every vertex is in every
Cg , g > k, then G is Kn , contrary to assumption.) Let C be the clique represented by
v. Replace the singleton {v1 } by {v} in C. This new collection of cliques (still called
C) is still a clique cover with a system of distinct representatives and has minimum
size. Choosing g so that v ∈
/ Cg , (Cg , C1 ) is an arc of CG(C, V).
II. d(C) ≥ n for C ∈ {Ck+1 , . . . , Cn }.
Actually, we show that either the theorem is true for G or d(C) ≥ n for C ∈
{Ck+1 , . . . , Cn }.
+
Consider a clique C = Ci , i > k. In the clique graph, d (C) = n − |C| and
−
d (C) = the number of cliques not containing vi
= n − the number of cliques containing vi
≥ n − (1 + n − |C|) = |C| − 1
by (MP). Thus, d(C) ≥ n − 1.
If for some C = Ci , d(C) = n − 1, then the number of cliques containing vi is
exactly 1 + n − |C|. (Note that |C| ≥ 2. If not, then k = n − 1 and i = n, so
−
d (C) = n − 1.) Let j = n − |C|. Let the cliques other than C that contain vi be
A1 , . . . , Aj . By (MP), there are vertices w1 , . . . , wj such that Al \C = {wl } for all
l. Now we consider the cliques other than C and A1 , . . . , Aj . Note that any clique
contained in C must be a singleton, since C was chosen to have minimum size.
Suppose there is a clique D that contains more than one of {w1 , . . . , wj }; without
loss of generality, say {w1 , . . . , wt } ⊆ D. Remove vi from A1 , . . . , At , and add vi to D.
This new collection of cliques still covers G, has a system of distinct representatives,
and has smaller size than C, which is a contradiction.
Suppose that D is a clique (not one of A1 , . . . , Aj ) containing one of {w1 , . . . , wj },
say, ws , and D is not a singleton. If we replace As by D ∪ As and D by its representative, we get a clique cover of G with a system of distinct representatives and the same
size as C. If we do this for each such D, we produce a clique cover, still called C for
convenience, consisting of C, A1 , . . . , Aj and |C|−1 ≥ 1 singletons. Moreover, we may
assume that each Am is represented by wm , and all of the singletons {x1 , . . . , x|C|−1 }
are in C.
Suppose that one of the singletons, without loss of generality, x1 , is not in some
A, without loss of generality, Aj . Then C, A1 , . . . , Aj , x1 ,. . . , x|C|−1 is a hamilton
circuit in CG(C, V).
Otherwise, suppose that every singleton is in every clique A. Split Aj into two
cliques. One, still called Aj for convenience, is Aj \{x1 }. The other, Aj+1 , contains wj
and x1 . Remove the singleton {x1 } from C. If we let x1 represent Aj+1 , then this new
collection of cliques is still a cover, still has a system of distinct representatives, and
has the same size as C. In the clique graph of this new collection, C, A1 , . . . , Aj , Aj+1 ,
132
DAVID R. GUICHARD
x2 , . . . , x|C|−1 is a hamilton circuit. (Note that if |C| = 2, there are no singletons left,
/ Aj+1 .)
but C, A1 , . . . , Aj , Aj+1 is a hamilton circuit, since vi ∈
Thus, we may assume that d(Ci ) ≥ n, i > k.
III. CG(C, V) is strongly connected.
Consider two distinct cliques, Ci and Cj .
If i ≤ k and j > k, then by definition of k, (Ci , Cj ) is an arc.
If i, j > k and (Ci , Cj ) is not an arc, then vj ∈ Ci . Let v ∈ Cj be such that
{vj , v} is in no clique other than Cj . Let C be the clique represented by v, so vj ∈
/ C.
Then (Ci , C) and (C, Cj ) are arcs.
Suppose i, j ≤ k. If G′ is not a complete graph, then C1 , . . . , Ck form a circuit,
so there is a path from Ci to Cj .
If G′ is the complete graph, the cliques C1 , . . . , Ck must consist of k − 1 singletons
and a copy of Kk . From (I) we know that C1 , . . . , Ck−1 are singletons and that for
some g > k, (Cg , C1 ) is an arc. Taken in order, C1 , . . . , Ck form a path, so if i < j,
there is a path from Ci to Cj . If i > j, we may use the path Ci , Cg , C1 , . . . , Cj .
Finally, suppose i > k and j ≤ k. If G′ is not a complete graph and C1 , . . . , Ck
are not all singletons, then (Ci , Cl ) is an arc for some l ≤ k, and since C1 , . . . , Ck
form a circuit, there is a path from Cl to Cj . If all of C1 , . . . , Ck are singletons, there
is a g > k and an l ≤ k such that (Cg , Cl ) is an arc; since there is a path from Ci to
Cg and from Cl to Cj , there is a path from Ci to Cj . If G′ is a complete graph, we
know there is a g > k such that (Cg , C1 ) is an arc; since there are paths from Ci to
Cg and C1 to Cj , we are done.
IV. Special cases.
Case 1. k = 0.
CG(C, V) satisfies the hypotheses of Theorem 4, so it is hamiltonian.
Case 2. k = 1.
+
By definition of k, this means that C1 is a singleton, so d (C1 ) = n−1. Previously,
−
we had guaranteed that there is some g > k such that (Cg , C1 ) is an arc, so d (C1 ) ≥ 1.
Now CG(C, V) satisfies the hypotheses of Theorem 4, so it is hamiltonian.
Case 3. All of C1 , . . . , Ck are singletons, and k > 1.
−
+
If i ≤ k, then d (Ci ) = n − 1 and d (Ci ) ≥ 1 because each of the other singletons
has an arc to Ci . Again, CG(C, V) satisfies the hypotheses of Theorem 4, so it is
hamiltonian.
Case 4. G′ is a complete graph.
Recall that this means that C1 , C2 , . . . , Ck−1 are all singletons, and Ck ∼
= Kk .
−
+
For i ≤ k − 1, d (Ci ) = n − 1. For 2 ≤ i ≤ k − 1, (Ci−1 , Ci ) is an arc, so d (Ci ) ≥ 1.
−
By (I), we know that there is some g > k such that (Cg , C1 ) is an arc, so d (C1 ) ≥ 1.
Thus, d(Ci ) ≥ n for i ≤ k − 1.
−
+
d (Ck ) = n − k and d (Ck ) ≥ (k − 1) + 1 = k. The “k − 1” is due to C1 , . . . , Ck−1 .
By the discussion in (I), there is some g > k such that (Cg , Ck ) is an arc, which
explains the “+1.” Thus, CG(C, V) satisfies the hypotheses of Theorem 4, so it is
hamiltonian.
V. The rest of the story.
Now we may assume that C1 , . . . , Ck form a circuit, and that k ≥ 2. Also, we may
assume that not all of C1 , . . . , Ck are singletons, so that for all g > k there is some i ≤ k
COMPETITION GRAPHS OF HAMILTONIAN DIGRAPHS
133
+
such that (Cg , Ci ) is an arc. Following Berge [1], we let Γ (v) = {w | (v, w) is an arc}
−
and Γ (v) = {w | (w, v) is an arc}.
Let X0 be the longest circuit in CG(C, V) that incorporates all of C1 , . . . , Ck in
that order (but not necessarily consecutively). Let m be the length of X0 ; m ≥ 2.
Denote the vertices in X0 by V0 , V1 , . . . , Vm−1 (in order around the circuit). If m = n,
we are done; for a contradiction, suppose that m < n. Let X1 , . . . , Xp be the strongly
connected components of CG(C) − X0 .
CLAIM. X1 contains a circuit of length |X1 | = q.
−
+
Suppose that V ∈ X1 . For all l, Vl ∈ Γ (V ) implies Vl+1 ∈
/ Γ (V ), for otherwise
X0 could be lengthened. (Note: Vm = V0 .) Hence,
+
−
|Γ (V ) ∩ X0 | ≤ |X0 | − |Γ (V ) ∩ X0 |.
+
−
For W ∈ Xj , j 6= 0, 1, W ∈ Γ (V ) implies W ∈
/ Γ (V ) because Xj is a strongly
connected component different from X1 . Hence, for j =
6 0, 1,
+
−
|Γ (V ) ∩ Xj | ≤ |Xj | − |Γ (V ) ∩ Xj |.
Since d(V ) ≥ n,
d(V ) =
p X
j=0
p
X
−
+
|Γ (V ) ∩ Xj | + |Γ (V ) ∩ Xj | ≥
|Xj | = n,
j=0
and so
−
+
|Γ (V )∩X1 |+|Γ (V )∩X1 |−|X1 | ≥ −
X
−
+
|Γ (V ) ∩ Xj | + |Γ (V ) ∩ Xj | − |Xj | ≥ 0.
j6=1
By Theorem 4, X1 is hamiltonian. Denote the vertices of X1 by W0 , W1 , . . . , Wq−1 ,
in order around the circuit. This proves the claim.
CLAIM. q < m.
If not, we can form a circuit by inserting C1 , . . . , Ck in order into X1 . (Pick any
V ∈ X1 ; then (V, Ci ) is an arc for some i ≤ k, and Ci , Ci+1 , . . . , Ck , C1 , . . . , Ci−1 may
be inserted immediately after V in X1 .) This forms a circuit containing C1 , . . . , Ck
that is longer than X0 , which is a contradiction. This proves the claim.
−
+
CLAIM. Suppose that Vi ∈ Γ (Ws ). Then Vi+1 , . . . , Vi+q are not in Γ (Ws−1 ).
(All subscript arithmetic wraps around as appropriate.)
+
For if Vi+j ∈ Γ (Ws−1 ), the circuit V0 , . . . , Vi , Ws , Ws+1 , . . . , Ws−1 , Vi+j , . . . ,
Vm−1 is longer than X0 . Suppose some of Vi+1 , . . . , Vi+j−1 are in {C1 , . . . , Ck }; by
definition of X0 , these vertices must be Cg , Cg+1 , . . . , Cg+h , for some g and h. (Note
that subscript arithmetic here wraps around at k, not n.) These may be inserted in
the new circuit immediately following Cg−1 ; if {Cg , . . . , Cg+1 } = {C1 , . . . , Ck }, then
{C1 , . . . , Ck } can be inserted anywhere, as in the previous claim. This produces a
circuit longer than X0 that contains all of C1 , . . . , Ck in order, which is a contradiction.
This proves the claim.
Now we show that for each Ws ,
−
+
|Γ (Ws ) ∩ X0 | + |Γ (Ws−1 ) ∩ X0 | ≤ m − q + 1.
+
Color the vertices of X0 as follows: if Vj ∈ Γ (Ws−1 ), color it red; otherwise, blue.
−
+
We know that both (Γ (Ws ) ∩ X0 ) and (Γ (Ws−1 ) ∩ X0 ) are nonempty (by properties
134
DAVID R. GUICHARD
of {C1 , . . . , Ck }), so there is at least one red vertex, and by the preceding paragraph
there is at least one sequence of q blue vertices. Thus, by Lemma 5,
+
−
|Γ (Ws ) ∩ X0 | + |Γ (Ws−1 ) ∩ X0 | ≤ nr + nb ≤ m − q + 1.
Finally, for our contradiction, we show that for some s,
+
−
|Γ (Ws ) ∩ X0 | + |Γ (Ws−1 ) ∩ X0 | ≥ m − q + 2.
For every W ∈ X1 , since d(W ) ≥ n,
−
+
|Γ (W ) ∩ X0 | + |Γ (W ) ∩ X0 | ≥ |X0 |
−
+
− |Γ (W ) ∩ X1 | + |Γ (W ) ∩ X1 | − |X1 |
X −
+
|Γ (W ) ∩ Xj | + |Γ (W ) ∩ Xj | − |Xj |
−
j6=0,1
≥ m − ((q − 1) + (q − 1) − q) − 0 = m − q + 2.
Now counting the arcs between X0 and X1 in two different ways, we have
q−1 X
s=0
q−1 X
−
+
−
+
|Γ (Ws ) ∩ X0 | + |Γ (Ws ) ∩ X0 |
|Γ (Ws ) ∩ X0 | + |Γ (Ws−1 ) ∩ X0 | =
s=0
≥ q(m − q + 2).
Thus, for at least one s, the desired inequality holds.
REFERENCES
[1] Claude Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1976.
[2] K. F. Fraughnaugh, J. R. Lundgren, S. K. Merz, J. S. Maybee, N. J. Pullman, Competition
graphs of strongly connected and Hamiltonian digraphs, SIAM J. Discrete. Math., 8 (1995),
pp. 179–185.