Choosability, Edge Choosability, and Total Choosability of Outerplane Graphs

Let χl (G), χ ′ l (G), χ ′′ l (G), and 1(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 and χl (G) = 2 if and only if G is bipartite with at most one cycle. (2) 1(G) ≤ χ ′ l (G) ≤ 1(G) + 1 and χ ′ l (G) = 1(G) + 1 if and only if G is an odd cycle. This proves the well-known list edge coloring conjecture for outerplane graphs. (3) χ ′′ l (G) = 1(G)+ 1 if 1(G) ≥ 4 and χ ′′ l (G) ≤ 5 if 1(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.