List-coloring the square of a subcubic graph

The square G2 of a graph G is the graph with the same vertex set as G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that for a planar graph G with maximum degree (G) = 3 we have (G2) 7. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of G2 equals the chromatic number of G2, that is l(G2) = (G2) for all G. If true, this conjecture (together with Thomassen's result) implies that every planar graph G with (G) = 3 satisfies l(G2) 7. We prove that every graph (not necessarily planar) with (G) = 3 other than the Petersen graph satisfies l(G2) 8 (and this is best possible). In addition, we show that if G is a planar graph with (G) = 3 and girth g(G) 7, then l(G2) 7. Dvor

• Type: Journal
• Year: 2008
• Where: JGT
• Authors: Daniel W. Cranston, Seog-Jin Kim
• Keywords: JGT 2008, Linear-time Coloring Algorithms, Planar Graph, Square G2