THE COMPETITION NUMBERS OF HAMMING GRAPHS WITH DIAMETER AT MOST THREE

- Journal title : Journal of the Korean Mathematical Society
- Volume 48, Issue 4, 2011, pp.691-702
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2011.48.4.691

Title & Authors

THE COMPETITION NUMBERS OF HAMMING GRAPHS WITH DIAMETER AT MOST THREE

Park, Bo-Ram; Sano, Yoshio;

Park, Bo-Ram; Sano, Yoshio;

Abstract

The competition graph of a digraph D is a graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs. In this paper, we compute the competition numbers of Hamming graphs with diameter at most three.

Keywords

competition graph;competition number;edge clique cover;Hamming graph;

Language

English

Cited by

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References

1.

H. H. Cho and S.-R. Kim, The competition number of a graph having exactly one hole, Discrete Math. 303 (2005), no. 1-3, 32-41.

2.

J. E. Cohen, Interval Graphs and Food Webs: a finding and a problem, Document 17696-PR, RAND Corporation, Santa Monica, CA, 1968.

3.

J. E. Cohen, Food Webs and Niche Space, Princeton University Press, Princeton, NJ, 1978.

4.

S.-R. Kim, The competition number and its variants, Quo vadis, graph theory?, 313-326, Ann. Discrete Math., 55, North-Holland, Amsterdam, 1993.

5.

S.-R. Kim, On competition graphs and competition numbers, Commun. Korean Math. Soc. 16 (2001), no. 1, 1-24.

6.

S.-R. Kim, Graphs with one hole and competition number one, J. Korean Math. Soc. 42 (2005), no. 6, 1251-1264.

7.

S.-R. Kim, J. Y. Lee, and Y. Sano, The competition number of a graph whose holes do not overlap much, Discrete Appl. Math. 158 (2010), no. 13, 1456-1460.

8.

S.-R. Kim, B. Park, and Y. Sano, The competition numbers of Johnson graphs, Discuss. Math. Graph Theory 30 (2010), 449-459.

9.

S.-R. Kim and F. S. Roberts, Competition numbers of graphs with a small number of triangles, Discrete Appl. Math. 78 (1997), no. 1-3, 153-162.

10.

S.-R. Kim and Y. Sano, The competition numbers of complete tripartite graphs, Discrete Appl. Math. 156 (2008), no. 18, 3522-3524.

11.

J. Y. Lee, S.-R. Kim, S.-J. Kim, and Y. Sano, The competition number of a graph with exactly two holes, Ars Combin. 95 (2010), 45-54.

12.

R. J. Opsut, On the computation of the competition number of a graph, SIAM J. Algebraic Discrete Methods 3 (1982), no. 4, 420-428.

13.

B. Park, S.-R. Kim, and Y. Sano, On competition numbers of complete multi- partite graphs with partite sets of equal size, preprint RIMS-1644 October 2008. (http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1644.pdf)

14.

B. Park, S.-R. Kim, and Y. Sano, The competition numbers of complete multipartite graphs and mutually orthog- onal Latin squares, Discrete Math. 309 (2009), no. 23-24, 6464-6469.

15.

B. Park and Y. Sano, The competition numbers of ternary Hamming graphs, Appl. Math. Lett. 24 (2011), 1608-1613.

16.

F. S. Roberts, Food webs, competition graphs, and the boxicity of ecological phase space, Theory and applications of graphs (Proc. Internat. Conf., Western Mich. Univ., Kala- mazoo, Mich., 1976), pp. 477-490. Lecture Notes in Math., Vol. 642, Springer, Berlin, 1978.

17.

Y. Sano, The competition numbers of regular polyhedra, Congr. Numer. 198 (2009), 211-219.