Some results on the injective chromatic number of graphs

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.