A proper s-coloring of an n-vertex graph is equitable if every color class has size n/s or n/s. A necessary condition to have an equitable s-coloring is that every vertex v appears in an independent set of size at least n/s. That is ₕ ₕ (₆) αᵥ n/s. Various authors showed that when G is a tree and s 3 this obvious necessary condition is also sufficient. Kierstead, Kostochka, and Xiang asked whether this result holds more generally for all outerplanar graphs. We show that the answer is No when s=3, but that the answer is Yes when s 6. The case s\4, 5\ remains open. We also prove an analogous result for planar graphs, with a necessary and sufficient hypothesis. Fix s 40. Let G be a planar graph, and let w₀, w₁ be its 2 vertices with largest degrees. If there exist disjoint independent sets I₀, I₁ such that |I₀|=n/s and |I₁| = (n+1) /s and w₀, w₁ I₀ I₁, then G has an equitable s-coloring.
Cranston et al. (Fri,) studied this question.