Half-integral flows in a planar graph with four holes

Abstract

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 s 1, …, s r , t 1, …, t r are vertices of G such that each pair { s i , t i } belongs to the boundary of some of I, J, K, O, and that the graph ( VG, EG∪ {{ s 1, t 1}, …, { s r , t r }}) is eulerian. We prove that if the multi(commodity)flow problem in G with unit demands on the values of flows from s i to t i ( i = 1, …, r) has a solution then it has a half-integral solution as well. In other words, there exist paths P 1 1, P 2 1, P 1 2, P 2 2, …, p 1 r , P 2 r in G such that each p j i connects s i and t i , 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 s i and t i , i = 1, …, r, provided that the corres-ponding multiflow problem has a solution, but this is, in general, false in case of four holes.)