A note on orientation and chromatic number of graphs

Abstract

Let D be any edge orientation of a graph G. We denote by $\Delta _k(D)$ the maximum value t for which there exists a directed path $v_1, \ldots , v_k$ such that $d^{out}(v_k)=t$, where $d^{out}(v_k)$ is the out-degree of $v_k$ in D. We first obtain some bounds for the chromatic number of G in terms of $\Delta _k(D)$ and then show a relationship between $\Delta _k(D)$ and vertex partitions of a graph into degenerate subgraphs.