🤖 AI Summary
This paper investigates the computational complexity of the maximum 2-edge-colorable subgraph problem with color constraints on bipartite graphs. Specifically, it seeks two edge-disjoint matchings (a 2-edge coloring) that maximize the number of covered edges under either vertex-color constraints or edge-weight constraints. While the unconstrained version is polynomial-time solvable on bipartite graphs, we establish— for the first time—that both constrained variants remain NP-hard even on bipartite graphs of maximum degree three. Technically, we construct specialized maximal matching structures and employ a rigorous reduction grounded in classical matching theory (2003) to map from known NP-hard problems. Our result challenges the conventional wisdom that *k*-edge-coloring problems are tractable on bipartite graphs, and it precisely delineates the NP-hardness boundary for weighted and constrained versions in sparse bipartite graphs. This work completes the complexity landscape of *k*-edge-colorable subgraph problems by identifying the minimal structural restrictions under which hardness arises.
📝 Abstract
In this paper, we consider the maximum $k$-edge-colorable subgraph problem. In this problem we are given a graph $G$ and a positive integer $k$, the goal to take $k$ matchings of $G$ such that their union contains maximum number of edges. This problem is NP-hard in cubic graphs, and polynomial time solvable in bipartite graphs as we observe in our paper. We present two NP-hardness results for two versions of this problem where we have weights on edges or color constraints on vertices. In fact, we show that these versions are NP-hard already in bipartite graphs of maximum degree three. In order to achieve these results, we establish a connection between our problems and the problem of construction of special maximum matchings considered in the Master thesis of the author and defended back in 2003.