On Maximum Cost $K_{t,t}$‐Free t‐Matchings of Bipartite Graphs

Abstract

Frank examined the maximum $K_{t,t}$-free $t$-matching problem of simple bipartite graphs. As the $C_6$-free $2$-matching problem is NP-hard (Geelen), this is a promising generalization of restricted $2$-matchings. Given an arbitrary family $\mathcal{T}$ of $K_{t,t}$-subgraphs of the underlying graph, a $\mathcal{T}$-free $t$-matching is a subgraph of maximum degree at most $t$ that contains no member of $\mathcal{T}$. We show that the maximum size $\mathcal{T}$-free $t$-matching problem also admits a nice min-max formula. Given an integer cost function on the edge-set which is vertex-induced on any member of $\mathcal{T}$, we also show an integer min-max formula for the maximum cost of $\mathcal{T}$-free $t$-matchings. As the maximum cost $C_4$-free 2-matching problem is NP-hard (Kira´ly), we cannot expect a nice characterization in general.