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2010 | 30 | 3 | 449-459
Tytuł artykułu

The competition numbers of Johnson graphs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The competition graph of a digraph D is a graph which has the same vertex set as D and has an edge between two distinct vertices 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 to characterize all graphs with given competition number k has been one of the important research problems in the study of competition graphs.
The Johnson graph J(n,d) has the vertex set ${v_X | X ∈ \binom{[n]}{d}$, where $\binom{[n]}{d}$ denotes the set of all d-subsets of an n-set [n] = {1,..., n}, and two vertices $v_{X₁}$ and $v_{X₂}$ are adjacent if and only if |X₁ ∩ X₂| = d - 1. In this paper, we study the edge clique number and the competition number of J(n,d). Especially we give the exact competition numbers of J(n,2) and J(n,3).
Wydawca
Rocznik
Tom
30
Numer
3
Strony
449-459
Opis fizyczny
Daty
wydano
2010
otrzymano
2009-04-14
poprawiono
2009-10-09
zaakceptowano
2009-10-10
Twórcy
  • Department of Mathematics Education, Seoul National University, Seoul 151-742, Korea
autor
  • Department of Mathematics Education, Seoul National University, Seoul 151-742, Korea
autor
  • Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Bibliografia
  • [1] H.H. Cho and S.-R. Kim, The competition number of a graph having exactly one hole, Discrete Math. 303 (2005) 32-41, doi: 10.1016/j.disc.2004.12.016.
  • [2] H.H. Cho, S.-R. Kim and Y. Nam, On the trees whose 2-step competition numbers are two, Ars Combin. 77 (2005) 129-142.
  • [3] J.E. Cohen, Interval graphs and food webs: a finding and a problem, Document 17696-PR, RAND Corporation (Santa Monica, CA, 1968).
  • [4] J.E. Cohen, Food webs and Niche space (Princeton University Press, Princeton, NJ, 1978).
  • [5] R.D. Dutton and R.C. Brigham, A characterization of competition graphs, Discrete Appl. Math. 6 (1983) 315-317, doi: 10.1016/0166-218X(83)90085-9.
  • [6] C. Godsil and G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics 207 (Springer-Verlag, 2001).
  • [7] S.G. Hartke, The elimination procedure for the phylogeny number, Ars Combin. 75 (2005) 297-311.
  • [8] S.G. Hartke, The elimination procedure for the competition number is not optimal, Discrete Appl. Math. 154 (2006) 1633-1639, doi: 10.1016/j.dam.2005.11.009.
  • [9] G.T. Helleloid, Connected triangle-free m-step competition graphs, Discrete Appl. Math. 145 (2005) 376-383, doi: 10.1016/j.dam.2004.06.010.
  • [10] W. Ho, The m-step, same-step, and any-step competition graphs, Discrete Appl. Math. 152 (2005) 159-175, doi: 10.1016/j.dam.2005.04.005.
  • [11] S.-R. Kim, The competition number and its variants, in: Quo Vadis, Graph Theory, (J. Gimbel, J.W. Kennedy, and L.V. Quintas, eds.), Annals of Discrete Mathematics 55 (North-Holland, Amsterdam, 1993) 313-326.
  • [12] S.-R. Kim, Graphs with one hole and competition number one, J. Korean Math. Soc. 42 (2005) 1251-1264, doi: 10.4134/JKMS.2005.42.6.1251.
  • [13] S.-R. Kim and F.S. Roberts, Competition numbers of graphs with a small number of triangles, Discrete Appl. Math. 78 (1997) 153-162, doi: 10.1016/S0166-218X(97)00026-7.
  • [14] S.-R. Kim and Y. Sano, The competition numbers of complete tripartite graphs, Discrete Appl. Math. 156 (2008) 3522-3524, doi: 10.1016/j.dam.2008.04.009.
  • [15] J.R. Lundgren, Food Webs, Competition Graphs, Competition-Common Enemy Graphs, and Niche Graphs, in: Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, IMH Volumes in Mathematics and Its Application 17 (Springer-Verlag, New York, 1989) 221-243.
  • [16] R.J. Opsut, On the computation of the competition number of a graph, SIAM J. Algebraic Discrete Methods 3 (1982) 420-428, doi: 10.1137/0603043.
  • [17] A. Raychaudhuri and F.S. Roberts, Generalized competition graphs and their applications, Methods of Operations Research, 49 (Anton Hain, Königstein, West Germany, 1985) 295-311.
  • [18] F.S. Roberts, Food webs, competition graphs, and the boxicity of ecological phase space, in: Theory and applications of graphs (Proc. Internat. Conf., Western Mich. Univ., Kalamazoo, Mich., 1976) (1978) 477-490.
  • [19] F.S. Roberts and L. Sheng, Phylogeny numbers for graphs with two triangles, Discrete Appl. Math. 103 (2000) 191-207, doi: 10.1016/S0166-218X(99)00209-7.
  • [20] M. Sonntag and H.-M. Teichert, Competition hypergraphs, Discrete Appl. Math. 143 (2004) 324-329, doi: 10.1016/j.dam.2004.02.010.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1506
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