Line game-perfect graphs (2301.00932v4)

Abstract: 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 $X\in{A,B}$ has the first move. $Y\in{A,B,-}$. If $Y\in{A,B}$, 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 $\chi_{[X,Y]}'(G)$ 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 $G$, [\chi_{[X,Y]}'(H)=\omega(L(H)),] where $\omega(L(H))$ denotes the clique number of the line graph of $H$. For each of the six possibilities $(X,Y)\in{A,B}\times{A,B,-}$, we characterise line $[X,Y]$-perfect graphs by forbidden (edge-induced) subgraphs and by explicit structural descriptions, respectively.