Bonds with parity constraints

Given a connected graph G = (V,E) and three even-sized subsets A1, A2, A3 of V , when does V have a partition (S1, S2) such that G[Si] is connected and |Si ∩Aj | is odd for all i = 1, 2 and j = 1, 2, 3? This problem arises in the area of integer flow theory and has theoretical interest in its own right. The special case when |A1| = |A2| = |A3| = 2 has been resolved by Chakravarti and Robertson, and the general problem can be rephrased as a problem on binary matroids that asks if a given triple of elements is contained in a circuit. The purpose of this paper is to present a complete solution to this problem based on a strengthening of Seymour’s theorem on triples in matroid circuits. Supported in part by NSF of China under Grant No. 10771209, and Chinese Academy of Sciences under Grant No. kjcx-yw-s7. Supported in part by NSF grant DMS-1001230 and NSA grant H98230-10-1-0186. Supported in part by NSA. Supported in part by the Research Grants Council of Hong Kong, and the Overseas and Hong Kong, Macau Young Scholars Collaborative Research Fund of the National Science Foundation of China.