# On Indicated Coloring of Some Classes of Graphs

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## Abstract

Indicated coloring is a type of game coloring 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 obtain structural characterization of \(\{P_5,K_4,Kite,Bull\}\)-free graphs and connected \(\{P_6,C_5, K_{1,3}\}\)-free graphs that contain an induced \(C_6\). Also, we prove that \(\{P_5,K_4,Kite,Bull\}\)-free graphs and connected \(\{P_6,C_5,\overline{P_5}, K_{1,3}\}\)-free graphs which contain an induced \(C_6\) are *k*-indicated colorable for all \(k\ge \chi (G)\). In addition, we show that, if \(m\ge 1\) and *G* is a bipartite graph, then \(\chi _i(\mathbb {K}[G](m,m,\ldots ,m))=\chi (\mathbb {K}[G](m,m,\ldots ,m))\). Further, we show that \(\mathbb {K}[C_5]\) is *k*-indicated colorable for all \(k\ge \chi (G)\) and as a consequence, we exhibit that \(\{P_2\cup P_3, C_4\}\)-free graphs, \(\{P_5,C_4\}\)-free graphs are *k*-indicated colorable for all \(k\ge \chi (G)\). This partially answers one of the questions which was raised by Grzesik (Discret Math 312:3467–3472, 2012).

## Keywords

Game chromatic number Indicated chromatic number \(P_5\)-free graphs## Mathematics Subject Classification

05C15 05C75## Notes

### Acknowledgements

The authors thank the referee for his or her very useful suggestions and comments which helped in the betterment of the paper. For the first author, this research was supported by the Council of Scientific and Industrial Research, Government of India, File no: 09/559(0096)/2012-EMR-I. For the second author, this research was supported by SERB DST Project, Government of India, File no: EMR/ 2016/007339. Also, for the third author, this research was supported by the UGC-Basic Scientific Research, Government of India, Login id: gokulnath.res@pondiuni.edu.in.

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