A note on $\mathtt{V}$-free $2$-matchings

TL;DR

This paper introduces the concept of V-free 2-matchings in bipartite graphs, verifies a related conjecture via hypergraph matching, and proves the NP-completeness of a covering problem involving these matchings.

Contribution

It defines V-free 2-matchings, verifies Liang's conjecture through hypergraph matching techniques, and establishes NP-completeness for a related covering decision problem.

Findings

Verification of Liang's conjecture using hypergraph matching.

Introduction of V-free 2-matchings in bipartite graphs.

NP-completeness of covering a bipartite graph's class with V-free 2-matchings.

Abstract

Motivated by a conjecture of Liang [Y.-C. Liang. {\em Anti-magic labeling of graphs}. PhD thesis, National Sun Yat-sen University, 2013.], we introduce a restricted path packing problem in bipartite graphs that we call a $\mathtt{V}$-free $2$-matching. We verify the conjecture through a weakening of the hypergraph matching problem. We close the paper by showing that it is NP-complete to decide whether one of the color classes of a bipartite graph can be covered by a $\mathtt{V}$-free $2$-matching.