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The Competition Numbers of Ternary Hamming Graphs View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applied Mathematics Letters 24 (2011) 1608–1613 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml The competition numbers of ternary Hamming graphs Boram Park a,∗, Yoshio Sano b a Department of Mathematics Education, Seoul National University, Seoul 151-742, Republic of Korea b Pohang Mathematics Institute, POSTECH, Pohang 790-784, Republic of Korea article info a b s t r a c t Article history: It is known to be a hard problem to compute the competition number k.G/ of a graph G in Received 29 June 2010 general. Park and Sano (in press) [16] gave the exact values of the competition numbers of Received in revised form 7 April 2011 Hamming graphs H.n; q/ if 1 ≤ n ≤ 3 or 1 ≤ q ≤ 2. In this paper, we give an explicit Accepted 8 April 2011 formula for the competition numbers of ternary Hamming graphs. ' 2011 Elsevier Ltd. All rights reserved. Keywords: Competition graph Competition number Edge clique cover Hamming graph 1. Introduction The notion of a competition graph was introduced by Cohen [1] as a means of determining the smallest dimension of ecological phase space. The competition graph C.D/ of a digraph D is a (simple undirected) graph which has the same vertex set as D and an edge between vertices u and v if and only if there is a vertex x in D such that .u; x/ and .v; x/ are arcs of D. For any graph G; G together with sufficiently many isolated vertices is the competition graph of an acyclic digraph. Roberts [2] defined the competition number k.G/ of a graph G to be the smallest number k such that G together with k isolated vertices is the competition graph of an acyclic digraph. Opsut [3] showed that the computation of the competition number of a graph is an NP-hard problem (see [4] for graphs whose competition numbers are known). It has been one of the important research problems in the study of competition graphs to compute the competition numbers for various graph classes (see [5–15], for recent research). For some special graph families, we have explicit formulas for computing competition numbers. For example, if G is a chordal graph without isolated vertices then k.G/ D 1, and if G is a nontrivial triangle-free connected graph then k.G/ D jE.G/j − jV .G/j C 2 (see [2]). In this paper, we study the competition numbers of Hamming graphs. For a positive integer q, we denote a q-set f1; 2 :::; qg by TqU. Also we denote the set of the n-tuple over TqU by TqUn. For positive integers n and q, the Hamming graph n H.n; q/ is the graph which has the vertex set TqU and in which two vertices x and y are adjacent if dH .x; y/ D 1, where n n dH VTqU × TqU ! Z is the Hamming distance defined by dH .x; y/ VD jfi 2 TnU j xi 6D yigj. Note that the diameter of the Hamming graph H.n; q/ is equal to n if q ≥ 2 and that the number of edges of the Hamming graph H.n; q/ is equal to 1 − n ≤ ≤ 2 n.q 1/q . Park and Sano [16] gave the exact values of the competition numbers of Hamming graphs H.n; q/ if 1 n 3 or 1 ≤ q ≤ 2 as follows. For n ≥ 1; H.n; 1/ has no edge and so k.H.n; 1// D 0. For n ≥ 1; H.n; 2/ is an n-cube and k.H.n; 2// D .n − 2/2n−1 C 2. For q ≥ 2; H.1; q/ is complete graph and so k.H.1; q// D 1. Theorem 1.1 ([16]). For q ≥ 2, we have k.H.2; q// D 2. ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (B. Park), [email protected], [email protected] (Y. Sano). 0893-9659/$ – see front matter ' 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.04.012 B. Park, Y. Sano / Applied Mathematics Letters 24 (2011) 1608–1613 1609 Theorem 1.2 ([16]). For q ≥ 3, we have k.H.3; q// D 6. In this paper, we give the exact values of the competition numbers of ternary Hamming graphs H.n; 3/. Our main result is the following: Theorem 1.3. For n ≥ 3, we have − k.H.n; 3// D .n − 3/3n 1 C 6: 2. Preliminaries We use the following notation and terminology in this paper. For a digraph D, a sequence v1; v2; : : : ; vn of the vertices of D is called an acyclic ordering of D if .vi; vj/ 2 A.D/ implies i < j. It is well known that a digraph D is acyclic if and only if − there exists an acyclic ordering of D. For a digraph D and a vertex v of D, we define the in-neighborhood ND .v/ of v in D to be f 2 j 2 g − the set w V .D/ .w; v/ A.D/ , and a vertex in ND .v/ is called an in-neighbor of v in D. For a graph G and a nonnegative integer k, we denote by G [ Ik the graph such that V .G [ Ik/ D V .G/ [ Ik and E.G [ Ik/ D E.G/, where Ik is a set of k isolated vertices. For a clique S of a graph G and an edge e of G, we say e is covered by S if both of the endpoints of e are contained in S. An edge clique cover of a graph G is a family of cliques of G such that each edge of G is covered by some clique in the family. The edge clique cover number θE .G/ of a graph G is the minimum size of an edge clique cover of G. An edge clique cover of G is called a minimum edge clique cover of G if its size is equal to θE .G/. n n−1 Let πj VTqU !TqU be a map defined by .x1;:::; xj−1; xj; xjC1;:::; xn/ 7! .x1;:::; xj−1; xjC1;:::; xn/. For j 2 TnU and p 2 TqUn−1, we let VD −1 D f 2 T Un j D g Sj.p/ πj .p/ x q πj.x/ p : (1) Note that Sj.p/ is a clique of H.n; q/ with size q. Let n−1 F .n; q/ VD fSj.p/ j j 2 TnU; p 2 TqU g: (2) Then F .n; q/ is the family of maximal cliques of H.n; q/. Park and Sano [16] showed the following: Lemma 2.1 ([16]). The following hold: • Let n ≥ 2 and q ≥ 2, and let K be a clique of H.n; q/ with size at least 2. Then there exists a unique maximal clique S of H.n; q/ containing K. • The edge clique cover number of H.n; q/ is equal to nqn−1. • Any minimum edge clique cover of H.n; q/ consists of edge disjoint maximum cliques. • The family F .n; q/ defined by (2) is a minimum edge clique cover of H.n; q/. Now we present the following lemma: Lemma 2.2. Let G be a graph. Suppose that any edge of G is contained in exactly one maximal clique of G. Let F be the family of all maximal cliques of size at least two in G, and let k be an integer with k ≥ k.G/. Then there exists an acyclic digraph D satisfying the following: (a) The competition graph of D is G [ Ik. − 2 [ f;g (b) For any vertex v in D; ND .v/ F . (c) The number of vertices which have no in-neighbor in D is equal to k C jV .G/j − jF j. Proof. Let k be an integer such that k ≥ k.G/. By the definition of the competition number of a graph, there exists an acyclic digraph satisfying (a). Suppose that any acyclic digraphs satisfying (a) does not satisfy (b). Let D be an acyclic digraph which jf 2 j − D ;gj maximizes v V .D/ ND .v/ among all acyclic digraphs satisfying (a) but not (b). Since D does not satisfy (b), there ∗ − ∗ 62 [ f;g jf 2 j − D ;gj exists a vertex v in D such that ND .v / F . Since D maximizes v V .D/ ND .v/ , we may assume that j − j 6D 2 ND .v/ 1 for any v V .D/. By the assumption that any edge of G is contained in exactly one maximal clique of G, any − ∗ clique of size at least two is contained in a unique maximal clique in G. Since ND .v / is a clique of size at least two, there 2 − ∗ exists a unique maximal clique S F containing ND .v /. Let σ VD .v1; v2; : : : ; vjV .D/j/ be an acyclic ordering of D, i.e., if .vi; vj/ 2 A.D/ then i < j. Let vi and vj be two vertices of S whose indices are largest in the acyclic ordering σ . Since vi and vj are adjacent in G, there is a vertex vl 2 V .D/ such that .vi; vl/ and .vj; vl/ are arcs of D. Then we define a digraph D1 by V .D1/ D V .D/ and D n f ∗ j 2 − ∗ g [ f j 2 g A.D1/ .A.D/ .x; v / x ND .v / / .x; vl/ x S : Then the digraph D1 is acyclic, since the index l of vl is larger than the index of any vertex in S.
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