Group chromatic number of planar graphs of girth at least 4

Jeager et al introduced a concept of group connectivity as an 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 paper 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.