turkmath.org

Let G be a graph. For a real number c, a c-sum flow of G is an assignment of non-zero real numbers to the edges of G such that the sum of values of all edges incident with each vertex is c. Let k be a natural number. A c-sum k-flow is a c-sum flow with values from the set {±1, . . . , ±(k - 1)}. In this talk, we present known results on c-sum k-flows of graphs and propose several conjectures. The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. Nowhere-zero flows in directed graphs are important because they have nice relations with four color theorem. Here, we provide a bridge between nowhere-zero flows in directed graphs and 0-sum flows in undirected graphs.