Resolution of indecomposable integral flows on signed graphs

Resolution of indecomposable integral flows on signed graphs

It is well-known that each nonnegative integral flow of a directed graph can be decomposed into a sum of nonnegative graph circuit flows, which cannot be further decomposed into nonnegative integral sub-flows. This is equivalent to saying that indecomposable flows of graphs are those graph circuit flows. Turning from graphs to signed graphs, the indecomposable flows are much richer than that of...

Classification of Conformally Indecomposable Integral Flows on Signed Graphs

A conformally indecomposable flow f on a signed graph Σ is a nonzero integral flow that cannot be decomposed into f = f1 + f2, where f1, f2 are nonzero integral flows having the same sign (both ≥ 0 or both ≤ 0) at every edge. This paper is to classify at integer scale conformally indecomposable flows into characteristic vectors of Eulerian cycle-trees — a class of signed graphs having a kind of...

Resolution of Irreducible Integral Flows on a Signed Graph

We completely describe the structure of irreducible integral flows on a signed graph by lifting them to the signed double covering graph. A (real-valued) flow (sometimes also called a circulation) on a graph or a signed graph (a graph with signed edges) is a real-valued function on oriented edges, f : ~ E → R, such that the net inflow to any vertex is zero. An integral flow is a flow whose valu...

Indecomposable Graphs and Signed Domination Numbers

Nowhere-zero flows on signed regular graphs

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...