Zero-Sum Flow Numbers of Regular Graphs
نویسندگان
چکیده
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 nonzero 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. We define the zero-sum flow number of G as the least integer k for which G admitting a zero-sum k-flow. In this paper, among others we calculate the zero-sum flow numbers for regular graphs and also the zero-sum flow numbers for Cartesian products of regular graphs with paths.
منابع مشابه
Zero-Sum Flows in Regular Graphs
For an undirected graph G, a zero-sum flow is an assignment of non-zero real numbers to the edges, such that the sum of the values of all edges incident with each vertex is zero. It has been conjectured that if a graph G has a zero-sum flow, then it has a zero-sum 6-flow. We prove this conjecture and Bouchet’s Conjecture for bidirected graphs are equivalent. Among other results it is shown that...
متن کامل0-Sum and 1-Sum Flows in Regular Graphs
Let G be a graph. Assume that l and k are two natural numbers. An l-sum flow on a graph G is an assignment of non-zero real numbers to the edges of G such that for every vertex v of G the sum of values of all edges incident with v equals l. An l-sum k-flow is an l-sum flow with values from the set {±1, . . . ,±(k − 1)}. Recently, it was proved that for every r, r > 3, r 6= 5, every r-regular gr...
متن کاملA Note on Zero-Sum 5-Flows in Regular Graphs
Let G be a graph. A zero-sum flow of G is an assignment of non-zero real numbers to the edges such that the sum of the values of all edges incident with each vertex is zero. Let k be a natural number. A zero-sum k-flow is a flow with values from the set {±1, . . . ,±(k − 1)}. It has been conjectured that every r-regular graph, r ≥ 3, admits a zero-sum 5-flow. In this paper we give an affirmativ...
متن کاملZero-Sum Magic Labelings and Null Sets of Regular Graphs
For every h ∈ N, a graph G with the vertex set V (G) and the edge set E(G) is said to be h-magic if there exists a labeling l : E(G) → Zh\{0} such that the induced vertex labeling s : V (G) → Zh, defined by s(v) = ∑ uv∈E(G) l(uv) is a constant map. When this constant is zero, we say that G admits a zero-sum h-magic labeling. The null set of a graph G, denoted by N(G), is the set of all natural ...
متن کاملOn 1-sum flows in undirected graphs
Let G = (V,E) be a simple undirected graph. For a given set L ⊂ R, a function ω : E −→ L is called an L-flow. Given a vector γ ∈ R , we say that ω is a γ-L-flow if for each v ∈ V , the sum of the values on the edges incident to v is γ(v). If γ(v) = c, for all v ∈ V , then the γ-L-flow is called a c-sum L-flow. In this paper we study the existence of γ-L-flows for various choices of sets L of re...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012