Paper

Line game-perfect graphs

Wai Lam Fong, Stephan, Dominique Andres

2024

Overview

The \\([X,Y]\\)-edge colouring game is played with a set of \\(k\\) colours on a graph \\(G\\) with initially uncoloured edges by two players, Alice (A) and Bob (B). The players move alternately. Player \\(Xın\\A,B\\\) has the first move. \\(Yın\\A,B,-\\\). If \\(Yın\\A,B\\\), then only player \\(Y\\) may skip any move, otherwise skipping is not allowed for any player. A move consists of colouring an uncoloured edge with one of the \\(k\\) colours such that adjacent edges have distinct colours. When no more moves are possible, the game ends. If every edge is coloured in the end, Alice wins; otherwise, Bob wins. The \\([X,Y]\\)-game chromatic index \\(_[X,Y]'(G)\\) is the smallest nonnegative integer \\(k\\) such that Alice has a winning strategy for the \\([X,Y]\\)-edge colouring game played on \\(G\\) with \\(k\\) colours. The graph \\(G\\) is called line \\([X,Y]\\)-perfect if, for any edge-induced subgraph \\(H\\) of \\(G\\), \\[_[X,Y]'(H)=(L(H)),\\] where \\((L(H))\\) denotes the clique number of the line graph of \\(H\\). For each of the six possibilities \\((X,Y)ın\\A,B\\\A,B,-\\\), we characterise line \\([X,Y]\\)-perfect graphs by forbidden (edge-induced) subgraphs and by explicit structural descriptions, respectively.

• Publisher: Cornell University Library, arXiv.org
• Subject: Games / Graph coloring / Graph theory / Graphs / Players