Equitable List Coloring of Graphs with Bounded Degree

A graph G is equitably k ‐choosable if for every k ‐list assignment L there exists an L ‐coloring of G such that every color class has at most ⌈ | G | / k ⌉ vertices. We prove results toward the conjecture that every graph with maximum degree at most r is equitably ( r + 1 ) ‐choosable. In particular, we confirm the conjecture for r ≤ 7 and show that every graph with maximum degree at most r and at least r 3 vertices is equitably ( r + 2 ) ‐choosable. Our proofs yield polynomial algorithms for corresponding equitable list colorings.