Nowhere-zero 4-flow in almost Petersen-minor free graphs

AbstractTutte [W.T. Tutte, On the algebraic theory of graph colorings, J. Combin. Theory 1 (1966) 15–20] conjectured that every bridgeless Petersen-minor free graph admits a nowhere-zero 4-flow. Let (P10)μ̄ be the graph obtained from the Petersen graph by contracting μ edges from a perfect matching. In this paper we prove that every bridgeless (P10)3̄-minor free graph admits a nowhere-zero 4-flow

Z3-connectivity of 4-edge-connected 2-triangular graphs

AbstractA graph G is k-triangular if each edge of G is in at least k triangles. It is conjectured that every 4-edge-connected 1-triangular graph admits a nowhere-zero Z3-flow. However, it has been proved that not all such graphs are Z3-connected. In this paper, we show that every 4-edge-connected 2-triangular graph is Z3-connected. The result is best possible. This result provides evidence to support the Z3-connectivity conjecture by Jaeger et al that every 5-edge-connected graph is Z3-connected

Integer flow and Petersen minor

Tutte [45] conjectured that every bridgeless Petersen-minor free graph admits a nowhere-zero 4-flow. Let P10 m&d1; ) be the graph obtained from the Petersen graph by contracting mu edges from a perfect matching. In chapter 1 we prove that every bridgeless P10 3&d1; -minor free graph admits a nowhere-zero 4-flow.;Walton and Welsh [48] proved that if a coloopless regular matroid M does not have a minor in {lcub}M(K 3,3), M*(K5){rcub}, then M admits a nowhere zero 4-flow. Lai et al [27] proved that if M does not have a minor in {lcub}M( K5), M*(K5){rcub}, then M admits a nowhere zero 4-flow. We prove in chapter 2 that if a coloopless regular matroid M does not have a minor in MP10 3&d1;, M*K5 , then M admits a nowhere zero 4-flow. This result implies Walton and Welsh [48] and Lai et al [27].;The odd-edge-connectivity of a graph G, denoted by lambda o(G), is the size of the smallest odd edge-cut of G. In chapter 3, some methods are developed to deal with small even edge-cuts and therefore, extending some earlier results from edge-connectivity to odd-edge-connectivity. One of the main results in chapter 3 solves an open problem that every odd-(2k + 1)-edge-connected graph has k edge-disjoint parity subgraphs. Another main theorem in the chapter generalizes an earlier result by Galluccio and Goddyn (Combinatorica 2002) that the flow index of every odd-7-edge-connected graph is strictly less than 4. It is also proved in this paper if lambda o(G) ≥ 4log2 &vbm0;VG&vbm0; , then G admits a nowhere-zero 3-flow which is a partial result to the weak 3-flow conjecture by Jaeger and improves an earlier result by Lai and Zhang[24]

On Nowhere Zero 4-Flows in Regular Matroids

Walton and Welsh proved that if a co-loopless regular matroid M does not have a minor in {M(K(3,3)),M∗(K5)}, then M admits a nowhere zero 4-flow. Lai, Li and Poon proved that if M does not have a minor in {M(K5),M∗(K5)}, then M admits a nowhere zero 4-flow. We prove that if a co-loopless regular matroid M does not have a minor in {M((P10)¯3 ),M∗(K5)}, then M admits a nowhere zero 4-flow where (P10)¯3 is the graph obtained from the Petersen graph P10by contracting 3 edges of a perfect matching. As both M(K3,3) and M(K5) are contractions of M((P10)¯3), our result extends the results of Walton and Welsh and Lai, Li and Poon

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

Degree sequence and supereulerian graphs

A sequence d = (d1, d2, · · · , dn) is graphic if there is a simple graph G with degree sequence d, and such a graph G is called a realization of d. A graphic sequence d is linehamiltonian if d has a realization G such that L(G) is hamiltonian, and is supereulerian if d has a realization G with a spanning eulerian subgraph. In this paper, it is proved that a nonincreasing graphic sequence d = (d1, d2, · · · , dn) has a supereulerian realization if and only if dn ≥ 2 and that d is line-hamiltonian if and only if either d1 = n − 1, or di=1 di ≤ ∑ dj≥2 (dj − 2)