Equitable defective coloring of sparse planar graphs

A graph has an equitable, defective kk-coloring (an ED-kk-coloring) if there is a kk-coloring of V(G)V(G) that is defective (every vertex shares the same color with at most one neighbor) and equitable (the sizes of all color classes differ by at most one). A graph may have an ED-kk-coloring, but no ED-(k+1)(k+1)-coloring. In this paper, we prove that planar graphs with minimum degree at least 22 and girth at least 1010 are ED-kk-colorable for any integer k≥3k≥3. The proof uses the method of discharging. We are able to simplify the normally lengthy task of enumerating forbidden substructures by using Hall’s Theorem, an unusual approach.