Indicated coloring of graphs

On Indicated Coloring of Some Classes of Graphs

“…In addition, we show that K[C 5 ] is k-indicated colorable for all k ≥ χ(G) and as a consequence, we exhibit that {P 2 ∪ P 3 , C 4 }-free graphs, {P 5 , C 4 }-free graphs are k-indicated colorable for all k ≥ χ(G). This partially answers one of the questions which was raised by A. Grzesik in [8].…”

“…See for instance, [9,15,18,19]. The idea of indicated coloring was introduced by A. Grzesik in [8] as a slight variant of the game coloring 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 aim of Ann as in game coloring is to achieve a proper coloring of the whole graph G, while Ben tries to "block" some vertex.…”

Indicated Coloring of Complete Expansion and Lexicographic Product of Graphs

“…, m 5 ), {P 5 , C 4 }-free graphs, connected {P 5 , P 2 ∪ P 3 , P 5 , Dart}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 [6]. In addition, if G is a Bipartite graph or a {P 5 , K 3 }-free graph (or) a {P 5 , P aw}-free graph and H ∈ F, then we have shown that χ i (G[H]) = χ(G[H]).…”

“…In [3,6,11], if one closely observe the proof's of the families of graphs in F while showing that they are k-indicated colorable for every k ≥ χ(G), we can see that the winning strategy of Ann will be independent of the choice of k. Hence any graph in F is also a graph in H. Also one can observe that if G is a bipartite graph, then I[G] is also a bipartite graph. Thus For studying the indicated coloring of the lexicographic product of {P 5 , K 3 }-free or {P 5 , P aw}free graphs with a graph in H with indicated chromatic number equal to its chromatic number, let us first consider the indicated coloring of the complete expansion of the independent expansion of a graph G.…”

Indicated coloring of matroids

“…Clearly χ i (G) ≥ χ(G). Grzesik [6] proved that if χ(G) = 2, then χ i (G) = 2 and gave an example of a graph G with χ(G) = 3 and χ i (G) = 4. He also shows an upper bound χ i (G) ≤ 4χ(G) for random graphs, and conjectures that indicated chromatic number of a graph is bounded by a function of chromatic number.…”