Constant Sum Group Flows of Graphs

Abstract

As an analogous concept of a nowhere-zero flow for directed graphs, zero-sum flows and constant-sum flows are defined and studied in the literature. For an undirected graph, a zero-sum flow (constant-sum flow resp.) is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero (constant h resp.), and we call it a zero-sum k-flow (h-sum k-flow resp.) if the values of the edges are less than k. We extend these concepts to general constant-sum A-flow, where A is an Abelian group, and consider the case A=Zk the additive Abelian cyclic group of integer congruences modulo k with identity 0. In the literature, a graph is alternatively called Zk-magic if it admits a constant-sum Zk-flow, where the constant sum is called a magic sum or an index for short. We define the set of all possible magic sums such that G admits a constant-sum Zk-flow to be Ik(G) and call it the magic sum spectrum, or for short, the index set of G with respect to Zk. In this article, we study the general properties of the magic sum spectrum of graphs. We determine the magic sum spectrum of complete bipartite graphs Km,n for m≥n≥2 as the additive cyclic subgroups of Zk generated by kd, where d=gcd(m−n,k). Also, we show that every regular graph G with a perfect matching has a full magic sum spectrum, namely, Ik(G)=Zk for all k≥3. We characterize a 3-regular graph so that it admits a perfect matching if and only if it has a full magic sum spectrum, while an example is given for a 3-regular graph without a perfect matching which has no full magic sum spectrum. Another example is given for a 5-regular graph without a perfect matching, which, however, has a full magic sum spectrum. In particular, we completely determine the magic sum spectra for all regular graphs of even degree. As a byproduct, we verify a conjecture raised by Akbari et al., which claims that every connected 4k-regular graph of even order admits a 1-sum 4-flow. More open problems are included.