In 1983, Bouchet proposed a conjecture that every flow-admissible signed graph admits nowhere-zero 6-flow. himself proved such graphs admit 216-flows and Zýka further 30-flows. this paper we show 11-flow.

Here, we give a simple proof of this theorem without using linear programming. Proof. Since G is 6-edge-connected, by Tutte [5, Theorem 1] or NashWilliams [4, Theorem 1], let T1,T2,T3 be three edge disjoint spanning trees of G. Let Pi be a parity subgraph of Ti (for i=1,2 only). Now fixing some orientation of G. Since P1∪P2 and G\E(P1) are even graphs, let f1 be a nowhere-zero 2-flow with suppo...

A nowhere-zero k-flow on a graph G is an assignment of a direction and a non-zero integer in absolute value smaller than k to each edge of G in such a way that, for each vertex, the sum of incoming values equals the sum of outgoing values. Nowhere-zero flow problems evolved from flowcolouring duality to a theory which has a central role in graph theory. There are many important flow problems wh...

It was conjectured by Tutte that every 4-edge-connected graph admits a nowherezero 3-flow. In this paper, we give a complete characterization of graphs whose squares admit nowhere-zero 3-flows and thus confirm Tutte’s 3-flow conjecture for the family of squares of graphs.

In 1954, Tutte conjectured that every bridgeless graph has a nowherezero 5-flow. Let ω be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte’s conjecture is equivalent to its restriction to cubic graphs with ω ≥ 2. We show that if a cubic graph G has no edge cut with fewer than 5 2ω − 1 edges that separates two odd cycles of a minimum 2-factor of G, then G has a n...

Tutte [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. c © 20...