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GRAPHS WITH ONE HOLE AND COMPETITION NUMBER ONE

  • KIM SUH-RYUNG (Department of Mathematics Education Seoul National University)
  • Published : 2005.11.01

Abstract

Let D be an acyclic digraph. The competition graph of D has the same set of vertices 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. The competition number of a graph G, denoted by k(G), is the smallest number k such that G together with k isolated vertices is the competition graph of an acyclic digraph. It is known to be difficult to compute the competition number of a graph in general. Even characterizing the graphs with competition number one looks hard. In this paper, we continue the work done by Cho and Kim[3] to characterize the graphs with one hole and competition number one. We give a sufficient condition for a graph with one hole to have competition number one. This generates a huge class of graphs with one hole and competition number one. Then we completely characterize the graphs with one hole and competition number one that do not have a vertex adjacent to all the vertices of the hole. Also we show that deleting pendant vertices from a connected graph does not change the competition number of the original graph as long as the resulting graph is not trivial, and this allows us to construct infinitely many graph having the same competition number. Finally we pose an interesting open problem.

Keywords

competition number;chordal graph;chordless cycle;hole

References

  1. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North Holland, New York, 1976
  2. C. Cable, K. F. Jones, J. R. Lundgren, and S. Seager, Niche graphs, Discrete Appl. Math. 23 (1989), 231-241 https://doi.org/10.1016/0166-218X(89)90015-2
  3. H. H. Cho and S-R. Kim, The Competition Number of a Graph Having Exactly One Hole, to appear in Discrete Math
  4. J. E. Cohen, Interval Graphs and Food Webs: A Finding and a Problem, RAND Corporation Document 17696-PR, Santa Monica, CA, 1968
  5. M. B. Cozzens and F. S. Roberts, T-Colorings of Graphs and the Channel As-signment Problem, Congr. Numer. 25 (1982) 191-208
  6. G. A. Dirac, On Rigid Circuit Graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961), 71-76 https://doi.org/10.1007/BF02992776
  7. W. K. Hale, Frequency Assignment: Theory and Application, Proc. IEEE 68 (1980), 1497-1514 https://doi.org/10.1109/PROC.1980.11899
  8. P. C. Fishburn and W. V. Gehrlein, Niche numbers, J. Graph Theory 16 (1992), 131-139 https://doi.org/10.1002/jgt.3190160204
  9. F. Harary, S-R. Kim, and F. S. Roberts, Competition numbers as a generalization of Turan's theorem, J. Ramanujan Math. Soc. 5 (1990), 33-43
  10. 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 B. V., Amsterdam, the Netherlands, 1993, 313- 326
  11. 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, (F. S. Roberts, ed.), IMH Volumes in Mathematics and Its Application, Vol. 17, Springer-Verlag, New York, 1989, 221-243
  12. R. J. Opsut, On the Computation of the Competition Number of a Graph, SIAM J. Discrete Math. 3 (1982), 420-428 https://doi.org/10.1137/0603043
  13. R. J. Opsut and F. S. Roberts, On the Fleet Maintenance, Mobile Radio fre- quency, Task Assignment and Traffc phasing Problem, The Theory and Applications of Graphs, (G. Chartrand, Y. Alavi, D. L. Goldsmith, L. Lesniak-Foster, and D. R. Lick, eds.), Wiley, New York, 1981, 479-492
  14. F. S. Roberts, Food Webs, Competition Graphs, and the Boxicity of Ecological Phase Space, Theory and Applications of Graphs, (Y. Alavi and D. Lick, eds.), Springer Verlag, New York, 1978, 477-490
  15. F. S. Roberts, Graph Theory and Its Applications to Problems of Society, SIAM, Pennsylvania, 1978
  16. D. Scott, The competition-common enemy graph of a digraph, Discrete Appl. Math. 17 (1987), 269-280 https://doi.org/10.1016/0166-218X(87)90030-8
  17. S. Seager, The Double Competition Number of Some Triangle-Free Graphs, Discrete Appl. Math. 29 (1990), 265-269
  18. C. E. Shannon, The Zero Capacity of a Noisy Channel, IEEE Trans. Inform. Theory IT-2 (1956), 8-19
  19. H. H. Cho, S-R. Kim, and Y. Nam, The m-Step Competition Graph of a Digraph, Discrete Appl. Math. 105 (2000), 115-127 https://doi.org/10.1016/S0166-218X(00)00214-6
  20. S-R. Kim and F. S. Roberts, Competition numbers of graphs with a small number of triangles, Discrete Appl. Math. 78 (1997), 153-162 https://doi.org/10.1016/S0166-218X(97)00026-7

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