Half-integral flows in a planar graph with four holes Original Research Article

Suppose that G = (VG, EG) is a planar graph embedded in the euclidean plane, that I, J, K, O are four of its faces (called holes in G), that s1, …, sr, t1, …, tr are vertices of G such that each pair {si, ti} belongs to the boundary of some of I,J,K,O, and that the graph (VG, EG∪ {{s1, t1}, …, {sr, tr}}) is eulerian.