DMGT

Discussiones Mathematicae Graph Theory 24(3) (2004) 373-387
DOI: https://doi.org/10.7151/dmgt.1237

ANALOGUES OF CLIQUES FOR ORIENTED COLORING

William F. Klostermeyer

Department of Computer and Information Sciences
University of North Florida
Jacksonville, FL 32224-2669, U.S.A.

Gary MacGillivray

Department of Mathematics and Statistics
University of Victoria
Victoria, Canada

Abstract

We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.

Keywords: graph coloring, oriented coloring, clique,planar graph.

2000 Mathematics Subject Classification: 05C15, 05C20, 05C69.

References

[1]	P. Hell and K. Seyffarth, Largest planar graphs of diameter two and fixed maximum degree, Discrete Math. 111 (1993) 313-322, doi: 10.1016/0012-365X(93)90166-Q.
[2]	A. Kostochka, E. Sopena, and X. Zhu, Acyclic and oriented chromatic numbers of graphs, J. Graph Theory 24 (1997) 331-340, doi: 10.1002/(SICI)1097-0118(199704)24:4<331::AID-JGT5>3.0.CO;2-P.
[3]	J. Nesetril, A. Raspaud, and E. Sopena, Colorings and girth of oriented planar graphs, Discrete Math. 165/166 (1997) 519-530, doi: 10.1016/S0012-365X(96)00198-7.
[4]	A. Raspaud and E. Sopena, Good and semi-strong colorings of oriented planar graphs, Info. Proc. Letters 51 (1994) 171-174, doi: 10.1016/0020-0190(94)00088-3.
[5]	K. Seyffarth, Maximal planar graphs of diameter two, J. Graph Theory 13 (1989) 619-648, doi: 10.1002/jgt.3190130512.
[6]	E. Sopena, The chromatic number of oriented graphs, J. Graph Theory 25 (1997) 191-205, doi: 10.1002/(SICI)1097-0118(199707)25:3<191::AID-JGT3>3.0.CO;2-G.
[7]	E. Sopena, Oriented graph coloring, Discrete Math. 229 (2001) 359-369, doi: 10.1016/S0012-365X(00)00216-8.
[8]	E. Sopena, There exist oriented planar graphs with oriented chromatic number at least sixteen, Info. Proc. Letters 81 (2002) 309-312, doi: 10.1016/S0020-0190(01)00246-0.
[9]	D. West, Introduction to Graph Theory (Prentice Hall, Upper Saddle River, NJ, 2001) (2nd edition).

Received 15 October 2002
Revised 11 March 2004