A note on orientation and chromatic number of graphs

Let D be any edge orientation of a graph G. We denote by ${\mathrm{\Delta }}_k(D)$ the maximum value t for which there exists a directed path $v_1,\dots ,v_k$ 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 ${\mathrm{\Delta }}_k(D)$ and then show a relationship between ${\mathrm{\Delta }}_k(D)$ and vertex partitions of a graph into degenerate subgraphs.