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.

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