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- W3005287990 abstract "Indicated coloring is a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph $G$ (regardless of Ben's strategy) is called the indicated chromatic number of $G$, denoted by $chi_i(G)$. In this paper, we have shown that for any graphs $G$ and $H$, $G[H]$ is $k$-indicated colorable for all $kgeqmathrm{col}(G)mathrm{col}(H)$. Also, we have shown 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 have shown that for some particular families of graphs $G$ and $H$, $G[H]$ is $k$-indicated colorable for every $kgeq chi(G[H])$. This serves as a partial answer to one of the questions raised by A. Grzesik in cite{and}. In addition, if $G$ is a Bipartite graph or a ${P_5,K_3}$-free graph (or) a ${P_5,Paw}$-free graph and if $H$ is from the same families of graphs, then we have shown that $chi_i(G[H])=chi(G[H])$." @default.
- W3005287990 created "2020-02-14" @default.
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- W3005287990 date "2020-02-04" @default.
- W3005287990 modified "2023-10-16" @default.
- W3005287990 title "On indicated coloring of lexicographic product of graphs" @default.
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- W3005287990 doi "https://doi.org/10.48550/arxiv.2002.01957" @default.
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