A proof of Cunningham's conjecture on restricted subgraphs and jump systems

  • Authors:

  • Yusuke Kobayashi;JáCint Szabó;Kenjiro Takazawa

  • Affiliations:

  • Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan;MTA-ELTE Egerváry Research Group, Institute of Mathematics of the Eötvös University, Budapest, Hungary;Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

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

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Abstract

For an undirected graph and a fixed integer k, a 2-matching is said to be k-restricted if it has no cycle of length k or less. The problem of finding a maximum cardinality k-restricted 2-matching is polynomially solvable when k==5. On the other hand, the degree sequences of the k-restricted 2-matchings form a jump system for k==5, which is consistent with the polynomial solvability of the maximization problem. In 2002, Cunningham conjectured that the degree sequences of 4-restricted 2-matchings form a jump system and the maximum cardinality 4-restricted 2-matching can be found in polynomial time. In this paper, we show that the first conjecture is true, that is, the degree sequences of 4-restricted 2-matchings form a jump system. We also show that the maximum weight 4-restricted 2-matchings in a bipartite graph induce an M-concave function on the jump system if and only if the weight function is vertex-induced on every square. This result is also consistent with the polynomial solvability of the maximum weight 4-restricted 2-matching problem in bipartite graphs.