On Game Chromatic Vertex-Critical Graphs

Abstract

Several games that arise from graph coloring have been introduced and studied. Let \(\varphi \) denote a graph invariant that arises from such a game. If G is a graph and \(\varphi (G-x)\ne \varphi (G)=k\), \(k \ge 1\), holds true for every vertex \(x \in V(G)\), then G is called a k-\(\varphi \)-game-vertex-critical graph. We study the concept of \(\varphi \)-game-vertex-criticality for \(\varphi \in \{\chi _g, \chi _i, \chi _\mathrm{{ig}}^\mathrm{{A}}, \chi _\mathrm{{ig}}^\mathrm{{AB}}\}\), where \(\chi _g\) denotes the standard game chromatic number, \(\chi _i\) denotes the indicated game chromatic number and \(\chi _\mathrm{{ig}}^\mathrm{{A}}\), \(\chi _\mathrm{{ig}}^\mathrm{{AB}}\) denote two versions of the independence game chromatic number. Since the game chromatic number \(\varphi (G-x)\) can either decrease or increase with respect to \(\varphi (G)\), we distinguish between lower, upper and mixed vertex-criticality. We show that for \(\varphi \in \{\chi _g, \chi _\mathrm{{ig}}^\mathrm{{A}}, \chi _\mathrm{{ig}}^\mathrm{{AB}}\}\) the difference \(\varphi (G)-\varphi (G-x)\), \(x \in V(G)\), can be arbitrarily large. A characterization of 2-\(\varphi \)-game-vertex-critical and (connected) 3-\(\varphi \)-lower-game-vertex-critical graphs for all \(\varphi \in \{\chi _g, \chi _i, \chi _\mathrm{{ig}}^\mathrm{{A}}, \chi _\mathrm{{ig}}^\mathrm{{AB}}\}\) is given. It is shown that \(\chi _g\)-game-vertex-critical, \(\chi _\mathrm{{ig}}^\mathrm{{A}}\)-game-vertex-critical and \(\chi _\mathrm{{ig}}^\mathrm{{AB}}\)-game-vertex-critical graphs are not necessarily connected. However, it is also shown that \(\chi _i\)-lower-game-vertex-critical graphs are always connected.