On 1-sum flows in undirected graphs

Let G = (V,E) be a simple undirected graph. For a given set L ⊂ R, a function ω : E −→ L is called an L-flow. Given a vector γ ∈ R , we say that ω is a γ-L-flow if for each v ∈ V , the sum of the values on the edges incident to v is γ(v). If γ(v) = c, for all v ∈ V , then the γ-L-flow is called a c-sum L-flow. In this paper we study the existence of γ-L-flows for various choices of sets L of real numbers, with an emphasis on 1-sum flows. Given a natural k number, a c-sum k-flow is a c-sum flow with values from the set {±1, . . . ,±(k − 1)}. Let L be a subset of real numbers containing 0 and denote L∗ := L \ {0}. Answering a question from [4] we characterize which bipartite graphs admit a 1-sum R∗-flow or a 1-sum Z∗-flow. We also show that that every k-regular graph, with k either odd or congruent to 2 modulo 4, admits a 1-sum {−1, 0, 1}-flow.