Integer flows and subgraph covers

Abstract

Integer flows and subgraph covers are two important subjects in graph theory. They are closely related to the famous four color problem, which is equivalent to the integer 4-flow problem in planar graphs. A graph has an integer $k$-flow if for some orientation, there is a function from its edge set to an abelian group such that at each vertex the sum of the values on the edges into the vertex is equal to the sum of the values on the edges out of the vertex. Integer flows are related to several well-known problems in other fields of mathematics, such as lonely runner problem in combinatorics, diophantine approximation in number theory, view obstruction in geometry and additive basis in linear vector space. Four color problem is also equivalent to the problem of Eulerian subgraph covering of planar graphs: Every 2-edge-connected planar graph has three Eulerian subgraphs covering each edge precisely twice. The well-known Fulkerson conjecture claims that every 2-edge-connected graph (not necessary to be planar) has six Eulerian subgraphs covering each edge precisely four times. In this paper, we shall give a brief survey of the two subjects and other related problems.