Coloring of Graphs Avoiding Bicolored Paths of a Fixed Length.

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of graphs (Grünbaum in Isreal J Math 14(4):390–498, 1973) where bicolored copies of P 4 and cycles are not allowed, respectively. In this paper, we introduce a variation of these problems and study proper coloring of graphs not containing a bicolored path of a fixed length and provide general bounds for all graphs. A P k -coloring of an undirected graph G is a proper vertex coloring of G such that there is no bicolored copy of P k in G, and the minimum number of colors needed for a P k -coloring of G is called the P k -chromatic number of G, denoted by s k (G). We provide bounds on s k (G) for all graphs, in particular, proving that for any graph G with maximum degree d ≥ 2 , and k ≥ 4 , s k (G) ≤ ⌈ 6 10 d k - 1 k - 2 ⌉. Moreover, we find the exact values for the P k -chromatic number of the products of some cycles and paths for k = 5 , 6. [ABSTRACT FROM AUTHOR]