On the computational complexity of the bipartizing matching problem

Given a graph $$G=(V,E)$$ , an edge bipartization set of G is a subset $$E'\subseteq E(G)$$ such that  $$G'=(V,E{\setminus } E')$$ is bipartite. An edge bipartization set that is also a matching of G is called a bipartizing matching of G. Let  $${\mathscr {BM}}$$ be the family of all graphs admitting a bipartizing matching. Although every graph has an edge bipartization set, the problem of recognizing graphs G having bipartizing matchings ( $$G\in \mathscr {BM}$$ ) is challenging. A (k, d)-coloring of a graph G is a k-coloring of V(G) such that each vertex of G has at most d neighbors with the same color as itself. Clearly a (k, 0)-coloring is a proper vertex k-coloring of G and, for any  $$d>0$$ , the k-coloring is non-proper, also known as defective. A graph belongs to  $$\mathscr {BM}$$ if and only if it admits a (2, 1)-coloring. Lovasz (1966) proved that for any integer  $$k>0$$ , any graph of maximum degree  $$\varDelta $$ admits a  $$\left( k,\lfloor \varDelta /k \rfloor \right) $$ -coloring. In this paper, we show that it is NP-complete to determine whether a 3-colorable planar graph of maximum degree 4 belongs to $$\mathscr {BM}$$ , i.e., (2, 1)-colorable. Besides, we show that it is NP-complete to determine whether planar graphs of maximum degree 6 or 8 admit a (2, 2) or (2, 3)-coloring, respectively. Due to Lovasz (1966), our results are tight in the sense that on graphs with maximum degree three, five, and seven, there always exists a (2, 1), (2, 2), and (2, 3)-coloring, respectively. Finally, we present polynomial-time algorithms for particular graph classes, besides some remarks on the parameterized complexity of the problem of recognizing graphs in  $${\mathscr {BM}}$$ .