Chen–Lih–Wu equitable coloring conjecture

Prove that every connected graph G with maximum degree Δ(G) ≥ 2 has an equitable coloring with Δ(G) colors, except when G is a complete graph, an odd cycle, or a balanced complete bipartite graph with odd-sized parts.

Background

The Chen–Lih–Wu conjecture seeks to strengthen Brooks’s theorem by requiring a proper coloring with exactly Δ(G) colors whose color classes have sizes differing by at most one. The conjecture identifies complete graphs, odd cycles, and balanced complete bipartite graphs with odd-sized parts as the only exceptions.

The paper proves a degree-sequence realization result that provides support for the conjecture, but it does not establish the conjecture for every connected graph. The authors explicitly state that the conjecture remains unresolved, so it qualifies as an open problem under the stated inclusion criteria.