KAIST Discrete Math Seminar

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 Δ(G)=3 we have 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 χ_l(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 χ_l(G²)≤7. We prove that every graph (not necessarily planar) with Δ(G)=3 other than the Petersen graph satisfies χ_l(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 χ_l(G²)≤7. Dvořák, Škrekovski, and Tancer showed that if G is a planar graph with Δ(G)=3 and girth g(G)≥10, then χ_l(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 χ_l(G²)≤6. This is joint work with Daniel Cranston.