Graphs with k odd cycle lengths
Discrete Mathematics
Graph Theory With Applications
Graph Theory With Applications
Ore Condition and Nowhere-Zero 3-Flows
SIAM Journal on Discrete Mathematics
Circular flows of nearly Eulerian graphs and vertex-splitting
Journal of Graph Theory
Structure and Coloring of Graphs with Only Small Odd Cycles
SIAM Journal on Discrete Mathematics
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The relation between sizes of odd-cycles and chromatic numbers of graphs has been extensively studied. It was conjectured by Bollobas and Erdos (1990) [4] and proved by Gyarfas [A. Gyarfas, Graph with k odd cycle lengths, Discrete Math. 103 (1992) 41-48] that if a graphGhaskdifferent odd cycle sizes, then the chromatic number ofG, @g(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 lengths2k-1and2k+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. Rucky, R. Skrekovski, 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=3, if the size of every odd edge cut of a graph G is in {k,k+2,...,3k-2}, then G admits a nowhere-zero 3-flow if and only if G cannot be reduced to three well-defined families of graphs.