tailieunhanh - Báo cáo toán học: "On the Oriented Game Chromatic Number"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: On the Oriented Game Chromatic Number. | On the Oriented Game Chromatic Number J. Nesetrilt Dept. of Appl. Math. and Institute of Theoretical Computer Sciences ITI Charles University Prague Czech Republic nesetril@ E. Sopena LaBRI Universite Bordeaux 1 345 cours de la Liberation 33405 Talence Cedex France sopena@ Submitted March 6 2000 Accepted May 18 2000. Subject Classification 05C15 68R05. Keywords oriented graph coloring coloring games. Abstract We consider the oriented version of a coloring game introduced by Bodlaender On the complexity of some coloring games Internal. J. Found. Comput. Sci. 2 1991 133-147 . We prove that every oriented path has oriented game chromatic number at most 7 and this bound is tight that every oriented tree has oriented game chromatic number at most 19 and that there exists a constant t such that every oriented outerplanar graph has oriented game chromatic number at most t. 1 Introduction Let G be an undirected graph with vertex set V G and edge set E G and X be a set of colors. We consider the two-players game defined as follows. The two players are Alice and Bob and they play alternatively with Alice having the first move. Alice s goal is to provide a proper -coloring of G and Bob s goal is to prevent her from doing so. A move consists in choosing an uncolored vertex u and assigning it a color from the set X distinct from the colors previously assigned by either player to the neighbors of u. If after V G moves the graph is colored then Alice wins otherwise Bob wins. In other This work has been partially supported by the Barrande Grant 02887-WD and the NATO Collaborative Research Grant 97-1519. IPartially supported by the Project LN00A056 of the Czech Ministry of Education. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 no. 2 2001 R14 1 words Bob wins whenever an impass is reached before the whole graph is colored that is if at some intermediate step for every uncolored vertex u and every color a u has some neighbour with color .