On 1-sum flows in undirected graphs

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Let $G=(V, E)$ be a simple undirected graph. For a given set $L\subset \mathbb{R}$, a function $\omega: E \longrightarrow L$ is called an $L$-flow. Given a vector $\gamma \in \mathbb{R}^V$, $\omega$ is a $\gamma$-$L$-flow if for each $v\in V$, the sum of the values on the edges incident to $v$ is $\gamma(v)$. If $\gamma(v)=c$, for all $v\in V$, then the $\gamma$-$L$-flow is called a $c$-sum $L$-flow. In this paper, the existence of $\gamma$-$L$-flows for various choices of sets $L$ of real numbers is studied, with an emphasis on 1-sum flows. Let $L$ be a subset of real numbers containing $0$ and denote $L^*:=L\setminus \{0\}$. Answering a question from S. Akbari, M. Kano, and S. Zare. A generalization of $0$-sum flows in graphs. \emph{Linear Algebra Appl.}, 438:3629--3634, 2013.], the bipartite graphs which admit a $1$-sum $\mathbb{R}^*$-flow or a $1$-sum $\mathbb{Z}^*$-flow are characterized. It is also shown that every $k$-regular graph, with $k$ either odd or congruent to 2 modulo 4, admits a $1$-sum $\{-1, 0, 1\}$-flow.