The list version of the Borodin-Kostochka conjecture for graphs with large maximum degree

Abstract

Brooks' Theorem states that the chromatic number χ(G) of a graph G is at most its maximum degree Δ(G) when Δ(G)≥3 and its clique number ω(G) is at most Δ(G). Vizing strengthened this by proving that the list chromatic number χl(G) is also at most Δ(G) for these graphs. Borodin and Kostochka conjectured that χ(G)<Δ(G) for all graphs G with Δ(G)≥9 and ω(G)<Δ(G). Reed proved their conjecture for graphs G with Δ(G) sufficiently large. Here we prove the analogous result for list coloring: χl(G)<Δ(G) for all graphs G with Δ(G) sufficiently large and ω(G)<Δ(G).