Nowhere-zero 3-flows of graphs with prescribed sizes of odd edge cuts

The relation between sizes of odd-cycles and chromatic numbers of graphs has been extensively studied. It was conjectured by Bollobás and Erdős (1990)  [4] and proved by Gyárfás [A. Gyárfás, Graph with kk odd cycle lengths, Discrete Math. 103 (1992) 41–48] that if a graph  GGhas  kkdifferent odd cycle sizes, then the chromatic number of  GG, χ(G)≤2k+2χ(G)≤2k+2. Recently, Camacho and Schiermeyer [S.M. Camacho, I. Schiermeyer, Colorings of graphs with two consecutive odd cycle lengths, Discrete Math. 309 (2009) 4916–4919] proved that every graph containing only odd cycles of lengths  2k−12k−1and  2k+12k+1is 4-colorable. Wang [S.S. Wang, Structure and coloring of graphs with only small odd cycles, SIAM J. Discrete Math. 22 (2008) 1040–1072] characterizes all 3-colorable graphs whose only odd cycles are 3-cycles and 5-cycles; and Kaiser et al. [T. Kaiser, O. Rucký, R. Škrekovski, Graphs with odd cycle lengths 5 and 7 are 3-colorable, SIAM J. Discrete Math. 25 (2012) 1069–1088] proved that every graph with odd cycle lengths 5 and 7 is 3-colorable.In this paper, we investigate the dual version of this problem: the relation between integer flows and odd edge cuts (the integer flow problem is a dual version of the vertex coloring problem, and the dual of an odd-cycle is an odd-edge-cut). The main result in this paper shows that, for each odd integer k≥3k≥3, if the size of every odd edge cut of a graph GG is in {k,k+2,…,3k−2}{k,k+2,…,3k−2}, then GG admits a nowhere-zero 3-flow if and only if GG cannot be reduced to three well-defined families of graphs.