Bidirected graphs are a generalization of undirected graphs. For bidirected graphs, we can consider a problem whichi is a natural extension of the maximum weighted stable set problem for undirected graphs. Here we call this problem the generalized stable set problem. It is well known that the maximum weighted stable set problem is solvable in polynomial time for perfect undirected graphs. Perfe...

Let G = (V, E) be a digraph and f a mapping from E into an Abelian group A. Associated with f is its boundary aS, a mapping from V to A, defined by af(x) = c Dleavingxf(e)-Ceenteringx f(e). We say that G is A-connected if for every b: V-, A with Cx E V b(x) = 0 there is an f: E -+ A (0) with b = af: This concept is closely related to the theory of nowhere-zero flows and is being studied here in...

Jensen and Toft [10] conjectured that every 2-edge-connected graph without a K5minor has a nowhere zero 4-flow. Walton and Welsh [24] proved that if a coloopless regular matroid M does not have a minor in {M(K3,3),M(K5)}, then M admits a nowhere zero 4-flow. In this note, we prove that if a coloopless regular matroid M does not have a minor in {M(K5),M(K5)}, then M admits a nowhere zero 4-flow....

We study the flow spectrum S(G) and the integer flow spectrum S(G) of odd regular graphs. We show that there are signed graphs where the difference between the integer flow number and the flow number is greater than or equal to 1, disproving a conjecture of Raspaud and Zhu [7]. Let G be a (2t + 1)-regular graph. We show that if r ∈ S(G), then r = 2 + 1t or r ≥ 2 + 2 2t−1 . This result generaliz...

If G is undirected, then we say that it has a nowhere-zero Γ flow if the graph admits a nowhere-zero Γ flow after giving an orientation to all the edges. As we saw, if one orientation works then any does, since inverses exist in abelian groups. Definition 2 Let G be an undirected graph. For integer k ≥ 2, a nowhere-zero k-flow φ is an assignment φ : E → {1, . . . , k−1} such that for some orien...

For a connected graph G of order n ≥ 3, let f : E(G) → Zn be an edge labeling of G. The vertex labeling f ′ : V (G) → Zn induced by f is defined as f (u) = ∑ v∈N(u) f(uv), where the sum is computed in Zn. If f ′ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A modular edge-graceful labeling f of G is nowhere-zero if f(e) 6= 0 for all e ∈...

Tutte’s famous 5-flow conjecture asserts that every bridgeless graph has a nowhere-zero 5-flow. Seymour proved that every such graph has a nowhere-zero 6-flow. Here we give (two versions of) a new proof of Seymour’s Theorem. Both are roughly equal to Seymour’s in terms of complexity, but they offer an alternative perspective which we hope will be of value.