List‐coloring the square of a subcubic graph
Daniel W. Cranston, Seog‐Jin Kim · Journal of Graph Theory · 2007
Abstract The square G2 of a graph G is the graph with the same vertex set G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that every planar graph G with maximum degree Δ(G) = 3 satisfies χ(G2) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list‐chromatic number of G2 equals the chromatic number of G2, that is, χl(G2) = χ(G2) for all G. If true, this conjecture (together with Thomassen's result) implies that every planar graph G with Δ(G) = 3 satisfies χl(G2) ≤ 7. We prove that every connected graph (not necessarily planar) with Δ(G) = 3 other than the Petersen graph satisfies χl(G2) ≤8 (and this is best possible). In addition, we show that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 7, then χl(G2) ≤ 7. Dvořák, Škrekovski, and Tancer showed that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 10, then χl(G2) ≤6. We improve the girth bound to show that if G is a planar graph with Δ(G) = 3 and g(G) ≥ 9, then χl(G2) ≤ 6. All of our proofs can be easily translated into linear‐time coloring algorithms. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 65–87, 2008