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. Published by Elsevier B.V.
Williams, L., Vandenbussche, J., & Yu, G. (2012). Equitable defective coloring of sparse planar graphs. Discrete Mathematics, 312(5), 957-962.