Bonds with parity constraints

  • Authors:

  • Xujin Chen;Guoli Ding;Xingxing Yu;Wenan Zang

  • Affiliations:

  • Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, China;Mathematics Department, Louisiana State University, Baton Rouge, LA 70803, USA;School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA;Department of Mathematics, The University of Hong Kong, Hong Kong, China

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2012

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Abstract

Given a connected graph G=(V,E) and three even-sized subsets A"1, A"2, A"3 of V, when does V have a partition (S"1,S"2) such that G[S"i] is connected and |S"i@?A"j| 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 |A"1|=|A"2|=|A"3|=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.