Indicated Coloring of Cartesian Product of Graphs

Abstract

Indicated coloring of a graph G is a coloring in which there are two players Ann and Ben, Ann picks a vertex and Ben chooses a color for this vertex. The aim of Ann is to achieve a proper coloring of the whole graph G, while Ben tries to block the same. The smallest number of colors required for Ann to win the game on a graph G is called the indicated chromatic number of G and is denoted by $\chi _i(G).$ In this paper, we prove that $T\Box C_n, T\Box K_{n_1,n_2,\dots ,n_m}$ and $K_{n_1,n_2,\dots ,n_m}\Box C_m$ are k-indicated colorable for all k greater than or equal to the indicated chromatic number of their corresponding Cartesian product, where T is any tree. Also we prove that $\chi _i(K_{k_1,k_2,\dots ,k_m}\Box K_{l_1,l_2,\dots ,l_n}) =\chi (K_{k_1,k_2,\dots ,k_m}\Box K_{l_1,l_2,\dots ,l_n}).$ Finally we have given non-trivial examples of graphs G and H for which $\chi _i(G\Box H) >\chi (G\Box H).$