Zero-Sum Flow Numbers of Hexagonal Grids

Abstract

As an analogous concept of nowhere-zero flows for directed and bi-directed graphs, we consider zero-sum flows for undirected graphs in this article. For an undirected graph G, a zero-sum k -flow is an assignment of non-zero integers whose absolute values less than k to the edges, such that the sum of the values of all edges incident with each vertex is zero. Furthermore we generalize the notion via considering a combinatorial optimization problem, which is to calculate the zero-sum minimum flow number of a graph G, namely, the least integer k for which G may admit a zero-sum k-flow. The Zero-Sum 6-Flow Conjecture was raised by Akbari et al. in 2009: If a graph with a zero-sum flow, it admits a zero-sum 6-flow. It turns out that this conjecture was proved to be equivalent to the classical Bouchet 6-flow conjecture for bi-directed flows. In this paper, we study zero-sum minimum flow numbers of graphs induced from plane tiling by regular hexagons in an arbitrary way, namely, the hexagonal grid graphs. In particular we are able to verify the Zero-Sum 6-Flow Conjecture for the class of hexagonal grid graphs by determining the zero-sum flow number of any non-trivial hexagonal grid graph is 3 or 4. We further use the concept of dual graphs to specify classes of infinite families of hexagonal grid graphs with minimum flow numbers 3 and 4 respectively. Further open problems are included.