Some results on the injective chromatic number of graphs

Abstract

A k-coloring of a graph G=(V,E) is a mapping c:V→{1,2,…,k}. The coloring c is injective if, for every vertex v∈V, all the neighbors of v are assigned with distinct colors. The injective chromatic number χ _i (G) of G is the smallest k such that G has an injective k-coloring. In this paper, we prove that every K ₄-minor free graph G with maximum degree Δ≥1 has $\chi_{i}(G)\le \lceil \frac{3}{2}\Delta\rceil$. Moreover, some related results and open problems are given.