Even factors, jump systems, and discrete convexity

  • Authors:

  • Yusuke Kobayashi;Kenjiro Takazawa

  • Affiliations:

  • Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan;Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan and Research Institute for Mathematical Sciences, Kyoto Un ...

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2009

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Abstract

A jump system, which is a set of integer lattice points with an exchange property, is an extended concept of a matroid. Some combinatorial structures such as the degree sequences of the matchings in an undirected graph are known to form a jump system. On the other hand, the maximum even factor problem is a generalization of the maximum matching problem into digraphs. When the given digraph has a certain property called odd-cycle-symmetry, this problem is polynomially solvable. The main result of this paper is that the degree sequences of all even factors in a digraph form a jump system if and only if the digraph is odd-cycle-symmetric. Furthermore, as a generalization, we show that the weighted even factors induce an M-convex (M-concave) function on a constant-parity jump system. These results suggest that even factors are a natural generalization of matchings and the assumption of odd-cycle-symmetry of digraphs is essential.