Integer flows and Modulo Orientations

Tutte\u27s 3-flow conjecture (1970\u27s) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. A graph G admits a nowhere-zero 3-flow if and only if G has an orientation such that the out-degree equals the in-degree modulo 3 for every vertex. In the 1980ies Jaeger suggested some related conjectures. The generalized conjecture to modulo k-orientations, called circular flow conjecture, says that, for every odd natural number k, every (2k-2)-edge-connected graph has an orientation such that the out-degree equals the in-degree modulo k for every vertex. And the weaker conjecture he made, known as the weak 3-flow conjecture where he suggests that the constant 4 is replaced by any larger constant.;The weak version of the circular flow conjecture and the weak 3-flow conjecture are verified by Thomassen (JCTB 2012) recently. He proved that, for every odd natural number k, every (2k 2 + k)-edge-connected graph has an orientation such that the out-degree equals the in-degree modulo k for every vertex and for k = 3 the edge-connectivity 8 suffices. Those proofs are refined in this paper to give the same conclusions for 9 k-edge-connected graphs and for 6-edge-connected graphs when k = 3 respectively

On flows of graphs

Tutte\u27s 3-flow Conjecture, 4-flow Conjecture, and 5-flow Conjecture are among the most fascinating problems in graph theory. In this dissertation, we mainly focus on the nowhere-zero integer flow of graphs, the circular flow of graphs and the bidirected flow of graphs. We confirm Tutte\u27s 3-flow Conjecture for the family of squares of graphs and the family of triangularly connected graphs. In fact, we obtain much stronger results on this conjecture in terms of group connectivity and get the complete characterization of such graphs in those families which do not admit nowhere-zero 3-flows. For the circular flows of graphs, we establish some sufficient conditions for a graph to have circular flow index less than 4, which generalizes a new known result to a large family of graphs. For the Bidirected Flow Conjecture, we prove it to be true for 6-edge connected graphs

Flows and parity subgraphs of graphs with large odd-edge-connectivity

AbstractThe odd-edge-connectivity of a graph G is the size of the smallest odd edge cut of G. Tutte conjectured that every odd-5-edge-connected graph admits a nowhere-zero 3-flow. As a weak version of this famous conjecture, Jaeger conjectured that there is an integer k such that every k-edge-connected graph admits a nowhere-zero 3-flow. Jaeger [F. Jaeger, Flows and generalized coloring theorems in graphs, J. Combin. Theory Ser. B 26 (1979) 205–216] proved that every 4-edge-connected graph admits a nowhere-zero 4-flow. Galluccio and Goddyn [A. Galluccio, L.A. Goddyn, The circular flow number of a 6-edge-connected graph is less than four, Combinatorica 22 (2002) 455–459] proved that the flow index of every 6-edge-connected graph is strictly less than 4. This result is further strengthened in this paper that the flow index of every odd-7-edge-connected graph is strictly less than 4. The second main result in this paper solves an open problem that every odd-(2k+1)-edge-connected graph contains k edge-disjoint parity subgraphs. The third main theorem of this paper proves that if the odd-edge-connectivity of a graph G is at least 4⌈log2|V(G)|⌉+1, then G admits a nowhere-zero 3-flow. This result is a partial result to the weak 3-flow conjecture by Jaeger and improves an earlier result by Lai et al. The fourth main result of this paper proves that every odd-(4t+1)-edge-connected graph G has a circular (2t+1) even subgraph double cover. This result generalizes an earlier result of Jaeger