Zero-Sum Flow Numbers of Triangular Grids

Abstract

As an analogous concept of a nowhere-zero flow for directed graphs, we consider zero-sum flows for undirected graphs in this article. For an undirected graph G, a zero-sum flow is an assignment of non-zero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum k -flow if the values of edges are less than k. Note that from algebraic point of view finding such zero-sum flows is the same as finding nowhere zero vectors in the null space of the incidence matrix of the graph. We consider in more details a combinatorial optimization problem, by defining the zero-sum flow number of G as the least integer k for which G admitting a zero-sum k-flow. It is well known that grids are extremely useful in all areas of computer science. Previously we studied flow numbers over hexagonal grids and obtained the optimal upper bound. In this paper, with new techniques we give completely zero-sum flow numbers for certain classes of triangular grid graphs, namely, regular triangular grids, triangular belts, fans, and wheels, among other results. Open problems are listed in the last section.