Equitable defective coloring of sparse planar graphs

A graph has an equitable, defective k -coloring (an ED- k -coloring) if there is a k -coloring of 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- k -coloring, but no ED- ( k + 1 ) -coloring. In this paper, we prove that planar graphs with minimum degree at least 2 and girth at least 10 are ED- k -colorable for any integer k ≥ 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.