Tutte’s 3-flow conjecture asserts that every 4-edge-connected graph has a nowhere-zero 3-flow. In this note we prove that, if a regular graph of valency at least four admits a solvable group of automorphisms acting transitively on its vertex set and edge set, then it admits a nowhere-zero 3-flow.

The existence of an integral flow polynomial that counts nowhere-zero k-flows on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for flows on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero flows on a signed graph with values in an abelian group of odd order. Note to publisher: This paper does ...

Let G be a 2-edge-connected graph with o vertices of odd degree. It is well-known that one should (and can) add o 2 edges to G in order to obtain a graph which admits a nowhere-zero 2-flow. We prove that one can add to G a set of ≤ b o 4c, d2b o 5ce, and d2b o 7ce edges such that the resulting graph admits a nowhere-zero 3-flow, 4-flow, and 5-flow, respectively.

Whole genome shotgun assembly is the process of taking many short sequenced segments (reads) and reconstructing the genome from which they originated. We demonstrate how the technique of bidirected network flow can be used to explicitly model the double-stranded nature of DNA for genome assembly. By combining an algorithm for the Chinese Postman Problem on bidirected graphs with the constructio...

In this paper, some earlier results by Fleischner [H. Fleischner, Bipartizing matchings and Sabidussi’s compatibility conjecture, DiscreteMath. 244 (2002) 77–82] about edge-disjoint bipartizingmatchings of a cubic graphwith a dominating circuit are generalized for graphs without the assumption of the existence of a dominating circuit and 3-regularity. A pair of integer flows (D, f1) and (D, f2)...

It is shown that every (2p+ 1) log2(|V (G)|)-edge-connected graph G has a mod (2p+ 1)orientation, and that a (4p+ 1)-regular graph G has a mod (2p+ 1)-orientation if and only if V (G) has a partition (V , V −) such that ∀U ⊆ V (G), |∂G(U)| ≥ (2p+ 1)||U ∩ V | − |U ∩ V −||. These extend former results by Da Silva and Dahad on nowhere zero 3-flows of 5-regular graphs, and by Lai and Zhang on highl...

A vector Sd-flow is a flow whose flow values are vectors in Sd, where Sd is the set of all unit vectors in Rd+1. Jain [Open Problem Garden, http://www.openproblemgarden.org/op/unit vector flows (2007)] and Thomassen [J. Combin. Theory Ser. B., 108 (2014), pp. 81–91] proved that a graph has a vector S1-flow if it has a nowhere-zero integer 3-flow. Thomassen [J. Combin. Theory Ser. B., 108 (2014)...

Let G be a 2-edge-connected graph with o vertices of odd degree. It is well-known that one should (and can) add o 2 edges to G in order to obtain a graph which admits a nowhere-zero 2-flow. We prove that one can add to G a set of ≤ ⌊ o 4⌋, ⌈ 1 2⌊ o 5⌋⌉, and ⌈ 1 2⌊ o 7⌋⌉ edges such that the resulting graph admits a nowhere-zero 3-flow, 4-flow, and 5-flow,

Let G be a 2-edge-connected simple graph on n vertices, let A denote an abelian group with the identity element 0, and let D be an orientation of G. The boundary of a function f : E(G) → A is the function ∂ f : V (G) → A given by ∂ f (v) = ∑ e∈E+(v) f (e) − ∑ e∈E−(v) f (e), where E(v) is the set of edges with tail v and E(v) is the set of edges with head v. A graph G is A-connected if for every...