The Relaxed Game Chromatic Number of Graphs with Cut-Vertices

Abstract

In the $(r,d)$-relaxed colouring game, two players, Alice and Bob alternately colour the vertices of $G$, using colours from a set $\mathcal{C}$, with $|\mathcal{C}|=r$. A vertex $v$ can be coloured with $c,c\in \mathcal{C}$ if after colouring $v$, the subgraph induced by all vertices with $c$ has maximum degree at most $d$. Alice wins the game if all vertices of the graph are coloured. The $d$-relaxed game chromatic number of $G$, denoted by $\chi _g^{(d)}(G)$, is the least number $r$ for which Alice has a winning strategy for the $(r,d)$-relaxed colouring game on $G$. A $(r,d)$-relaxed edge-colouring game is the version of the $(r,d)$-relaxed colouring game which is played on edges a graph. The parameter associated with the $(r,d)$-relaxed edge-colouring game is called the $d$-relaxed game chromatic index and denoted by $^{(d)}\chi '_g(G)$. We consider the game on graphs containing cut-vertices. For $k\ge 2$ we define a class of graphs $\mathcal{H}_k =\{G|\mathrm{\;every \;block \;of\;} G \; \mathrm{has \;at \;most}\; k \;\mathrm{vertices}\}$. We find upper bounds on the $d$-relaxed game chromatic number of graphs from $\mathcal{H}_k\;(k\ge 5)$. Since the line graph of the forest with maximum degree $k$ belongs to $\mathcal{H}_k$, from results for $\mathcal{H}_k$ we obtain some new results for line graphs of forests, i.e., for the relaxed game chromatic index of forests. We prove that $^{(d)}\chi '_g(T)\le \Delta (T)+2-d$ for any forest $T$. Moreover, we show that for forests with large maximum degree we can derive better bounds. These results improve the upper bound on the relaxed game chromatic index of forests obtained in Dunn (Discrete Math 307:1767–1775, 2007). Furthermore, we determine minimum $d$ that guarantee Alice to win the $(2,d)$-relaxed edge-colouring game on forests.