Indicated Coloring of Complete Expansion and Lexicographic Product of Graphs

Abstract

Indicated coloring is a slight variant of the game coloring which was introduced by Grzesik [6]. In this paper, we show that for any graphs G and H, G[H] is k-indicated colorable for all $k\ge \mathrm {col}(G)\mathrm {col}(H)$. Also, we show that for any graph G and for some classes of graphs H with $\chi (H)=\chi _i(H)=\ell$, G[H] is k-indicated colorable if and only if $G[K_\ell ]$ is k-indicated colorable. As a consequence of this result we show that if $G\in \mathcal {G}=\Big \{$Chordal graphs, Cographs, $\{P_5,C_4\}$-free graphs, Complete multipartite graphs$\Big \}$ and $H\in \mathcal {F}=\Big \{$Bipartite graphs, Chordal graphs, Cographs, $\{P_5,K_3\}$-free graphs, $\{P_5,Paw\}$-free graphs, Complement of bipartite graphs, $\{P_5,K_4,Kite,Bull\}$-free graphs, connected $\{P_6,C_5,\overline{P_5},K_{1,3}\}$-free graphs which contain an induced $C_6$, $\mathbb {K}[C_5](m_1, m_2 ,\ldots ,m_5)$, $\{P_5,C_4\}$-free graphs, connected $\{P_5,\overline{P_2\cup P_3},\overline{P_5},$ $Dart\}$-free graphs which contain an induced $C_5\Big \}$, then G[H] is k-indicated colorable for every $k\ge \chi (G[H])$. This serves as a partial answer to one of the questions raised by Grzesik in [6].