An algorithm for finding a maximum [formula omitted]-matching excluding complete partite subgraphs

Abstract

For an integer t and a fixed graph H, we consider the problem of finding a maximum t-matching not containing H as a subgraph, which we call the H-free t-matching problem. This problem is a generalization of the problem of finding a maximum 2-matching with no short cycles, which has been well-studied as a natural relaxation of the Hamiltonian circuit problem. When H is a complete graph Kt+1 or a complete bipartite graph Kt,t, in 2010, Bérczi and Végh gave a polynomial-time algorithm for the H-free t-matching problem in simple graphs with maximum degree at most t+1. A main contribution of this paper is to extend this result to the case when H is a t-regular complete partite graph. We also show that the problem is NP-complete when H is a connected t-regular graph that is not complete partite. Since it is known that, for a connected t-regular graph H, the degree sequences of all H-free t-matchings in a graph form a jump system if and only if H is a complete partite graph, our results show that the polynomial-time solvability of the H-free t-matching problem is consistent with this condition.