Half-integral Flows in a Planar Graph with Four Holes

Suppose that G = (1/G, EG) is a planar graph embedded in the euclidean plane, that I, J, K, 0 are four of its faces (called holes in G), that sl, , s,, t, , , t, are vertices of G such that each pair {si, ti} belongs to the boundary of some of I,J, K, 0, and that the graph (I/G, EGu{ {.slr fl}, ___ ,{.r,, t,}}) is eulerian. We prove that if the multi(commodity)flow problem in G with unit demands on the values of flows from si to ti (i = 1, , r) has a solution then it has a haJf_integral solution as well. In other words, there exist paths Pi, PT, Pi, Pi,. , P,‘, Pf in G such that each Pi connects si and li, and each edge of G is covered at most twice by these paths. (It is known that in case of at most three holes there exist edge-disjoint paths connecting si and ti, i = 1, , r, provided that the corresponding multiflow problem has a solution. but this is, in general, false in case of four holes.)