Choosability, Edge Choosability, and Total Choosability of Outerplane Graphs

Abstract

Let χl(G),χl′ (G),χl′′ (G), and Δ(G) denote, respectively, the list chromatic number, the list chromatic index, the list total chromatic number, and the maximum degree of a non-trivial connected outerplane graph G. We prove the following results. (1) 2 ≤χl(G) ≤ 3 andχl (G) = 2 if and only if G is bipartite with at most one cycle. (2)Δ(G) ≤χl′(G) ≤Δ(G) + 1 andχl′ (G) =Δ(G) + 1 if and only if G is an odd cycle. This proves the well-known list edge coloring conjecture for outerplane graphs. (3)χl′′(G) =Δ(G) + 1 if Δ(G) ≥ 4 and χl′′(G) ≤ 5 if Δ(G) ≤ 3. This proves a conjecture of O. V. Borodin, A. V. Kostochka and D. R. Woodall, List edge and list total coloring of multigraphs, J. Comb. Theory B, 71 (1997), 184–204 for outerplane graphs.