On mod (2p +1)-orientations of graphs
Journal of Combinatorial Theory Series B
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Jeager 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