Injective (Δ + 2)-coloring of planar graphs with girth 5

An injective [Formula: see text]-coloring of a graph [Formula: see text], referring to a mapping [Formula: see text], such that [Formula: see text] when two distinct vertices [Formula: see text] and [Formula: see text] have a common neighbor. In this paper, by analyzing the structural properties of the minimal counterexamples and by using the discharging method, we prove that [Formula: see text] for planar graph [Formula: see text] with [Formula: see text] and [Formula: see text].