The cone of balanced subgraphs

In this paper we study a 2-color analog of the cycle cone of a graph. Suppose the edges of a graph are colored red and blue. A nonnegative real vector on the edges is said to be balanced if the red sum equals the blue sum at every vertex. A balanced subgraph is a subgraph whose characteristic vector is balanced (i.e., red degree equals blue degree at every vertex). By a sum (respectively, fractional sum) of cycles we mean a nonnegative integral (respectively, nonnegative rational) combination of characteristic vectors of cycles. Similarly, we define sum and fractional sum of balanced subgraphs. We show that a balanced sum of cycles is a fractional sum of balanced subgraphs.