Indicated list colouring game on graphs
Yangyan Gu, Yiting Jiang, Huan Zhou, Jialu Zhu, Xuding Zhu
TL;DR
The paper studies the indicated list-colouring game on graphs and proves a sharp structural dichotomy: a connected graph is not indicated degree-choosable exactly when it is an expanded Gallai-tree, enabling a linear-time decision algorithm. It also introduces IC-Brooks graphs, showing these are precisely the regular expanded Gallai-trees, and proves that for $r\le 3$ every $r$-regular expanded Gallai-tree is IC-Brooks, while for $r\ge 4$ there exist regular counterexamples. A key methodological contribution is the development of three graph operations and the construction of the constructible infeasible family $\mathcal{W}$, leading to the equivalence between infeasibility and constructibility. The results yield a Brooks-type theorem for indicated colouring and provide linear-time algorithms for recognizing indicated degree-choosability and IC-Brooks graphs in bounded-degree graphs. The work connects game-based chromatic concepts with Gallai-tree structure and clique-cut analysis, with implications for colorability thresholds and structural graph theory.
Abstract
Given a graph $G$ and a list assignment $L$ for $G$, the indicated $L$-colouring game on $G$ is played by two players: Ann and Ben. In each round, Ann chooses an uncoloured vertex $v$, and Ben colours $v$ with a colour from $L(v)$ that is not used by its coloured neighbours. If all vertices are coloured, then Ann wins the game. Otherwise after a finite number of rounds, there remains an uncoloured vertex $v$ such that all colours in $L(v)$ have been used by its coloured neighbours, Ben wins. We say $G$ is indicated $L$-colourable if Ann has a winning strategy for the indicated $L$-colouring game on $G$. For a mapping $g: V(G) \to \mathbb{N}$, we say $G$ is indicated $g$-choosable if $G$ is indicated $L$-colourable for every list assignment $L$ with $|L(v)| \ge g(v)$ for each vertex $v$, and $G$ is indicated degree-choosable if $G$ is indicated $g$-choosable for $g(v) =d_G(v)$ (the degree of $v$). This paper proves that a graph $G$ is not indicated degree-choosable if and only if $G$ is an expanded Gallai-tree - a graph whose maximal connected induced subgraphs with no clique-cut are complete graphs or blow-ups of odd cycles, along with a technical condition (see Definition \ref{def-egt}). This leads to a linear-time algorithm that determines if a graph is indicated degree-choosable. A connected graph $G$ is called an IC-Brooks graph if its indicated chromatic number equals $Δ(G)+1$. Every IC-Brooks graph is a regular expanded Gallai-tree. We show that if $r \le 3$, then every $r$-regular expanded Gallai-tree is an IC-Brooks graph. For $r \ge 4$, there are $r$-regular expanded Gallai-trees that are not IC-Brooks graphs. We give a characterization of IC-Brooks graphs, and present a linear-time algorithm that determines if a given graph of bounded maximum degree is an IC-Brooks graph.