Circular Zero-Sum r-Flows of Regular Graphs

A circular zero-sum flow for a graph G is a function $$f:E(G) \rightarrow {\mathbb {R}}{\setminus }\{0\}$$ such that for every vertex v, $$\sum _{e\in E_v}f(e)=0$$, where $$E_v$$ is the set of all edges incident with v. If for each edge e, $$1\le |f(e)| \le r-1$$, where $$r\ge 2$$ is a real number, then f is called a circular zero-sum r-flow. Also, if r is a positive integer and for each edge e, f(e) is an integer, then f is called a zero-sum r-flow. If G has a circular zero-sum flow, then the minimum $$r\ge 2$$ for which G has a circular zero-sum r-flow is called the circular zero-sum flow number of G and is denoted by $$\Phi _c(G)$$. Also, the minimum integer $$r\ge 2$$ for which G has a zero-sum r-flow is called the flow number for G and is denoted by $$\Phi (G)$$. In this paper, we investigate circular zero-sum r-flows of regular graphs. In particular, we show that if G is k-regular with m edges, then $$\Phi _c(G)=2$$ for even k and even m, $$\Phi _c(G)=1+\frac{k+2}{k-2}$$ for even k and odd m, and $$\Phi _c(G)\le 1+(\frac{k+1}{k-1})^2$$ for odd k. It was proved that for every k-regular graph G with $$k\ge 3$$, $$\Phi (G)\le 5$$. Here, using circular zero-sum flows, we present a new proof of this result when $$k \ne 5$$. Finally, we prove that a graph G has a circular zero-sum flow f such that for every edge e, $$l(e) \le f(e) \le u(e)$$, if and only if for every partition of V(G) into three subsets A, B, C, $$\begin{aligned} l(A,C)+2l(A) \le u(B,C)+2u(B), \end{aligned}$$where l(A, C) is the sum of values of l on the edges between A, C, and l(A) is the sum of values of l on the edges with both ends in A (the definitions of u(B, C) and u(B) are analogous).