On indicated coloring of lexicographic product of graphs

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 χ i ( G ) . In this paper, we show that for any graphs G and H , G [ H ] is k -indicated colorable for all k ≥ col ( G ) col ( H ) . Also, we show that for any graph G and for some classes of graphs H with χ ( H ) = χ i ( H ) = ℓ , G [ H ] is k -indicated colorable if and only if G [ K ℓ ] is k -indicated colorable. As a consequence of this result, we show that if G ∈ G = { Chordal graphs, Cographs, Complement of bipartite graphs, { P 5 , C 4 } -free graphs, Connected { P 6 , C 5 , P 5 ¯ , K 1 , 3 } -free graphs which contain an induced C 6 , Complete multipartite graphs } and H ∈ F = { Bipartite graphs, Chordal graphs, Cographs, { P 5 , K 3 } -free graphs, { P 5 , p a w } -free graphs, Complement of bipartite graphs, { P 5 , K 4 , k i t e , b u l l } -free graphs, Connected { P 6 , C 5 , P 5 ¯ , K 1 , 3 } -free graphs which contain an induced C 6 , { P 5 , C 4 } -free graphs, K [ C 5 ] ( m 1 , m 2 , … , m 5 ) , Connected { P 5 , P 2 ∪ P 3 ¯ , P 5 ¯ , d a r t } -free graphs which contain an induced C 5 } , then G [ H ] is k -indicated colorable for every k ≥ χ ( G [ H ] ) . This serves as a partial answer to one of the questions raised by A. Grzesik in Grzesik (2012). In addition, if G ∈ { Bipartite graphs, { P 5 , K 3 } -free graphs, { P 5 , p a w } -free graphs } and H ∈ F , then we show that χ i ( G [ H ] ) = χ ( G [ H ] ) .