Group chromatic number of planar graphs of girth at least 4

Abstract

AbstractJeager et al. introduced a concept of group connectivity as a generalization of nowhere zero flows and its dual concept group coloring, and conjectured that every 5‐edge connected graph is Z3‐connected. For planar graphs, this is equivalent to that every planar graph with girth at least 5 must have group chromatic number at most 3. In this article, we show that if G is a plane graph with girth at least 4 such that all 4 cycles are independent, every 4‐cycle is a facial cycle and the distance between every pair of a 4‐cycle and a 5‐cycle is at least 1, then the group chromatic number of G is at most 3. As a special case, we show that the conjecture above holds for planar graphs. We also prove that if G is a connected K3,3‐minor free graph with girth at least 5, then the group chromatic number is at most 3. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 51–72, 2006