Coloring of Graphs Avoiding Bicolored Paths of a Fixed Length

TL;DR

This paper introduces and analyzes a new graph coloring problem that avoids bicolored paths of fixed length, providing bounds on the minimum colors needed and exact values for certain graph products.

Contribution

It defines the $P_k$-coloring problem, establishes bounds on the $P_k$-chromatic number for all graphs, and computes exact values for specific cycle and path products.

Findings

Bounds on $s_k(G)$ for all graphs with maximum degree $d$

Exact $P_k$-chromatic numbers for products of cycles and paths for $k=5,6$

For $k eq 3$, $s_k(G)=O(d^{(k-1)/(k-2)})$

Abstract

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 (Gr\"unbaum, 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 $d\geq 2,$ and $k\geq4,$ $s_k(G)=O(d^{\frac{k-1}{k-2}}).$ Moreover, we find the exact values for the $P_k$-chromatic number of the products of some cycles and paths for $k=5,6.$