Equistarable bipartite graphs

TL;DR

This paper characterizes equistarable bipartite graphs, linking them to matchings and the triangle condition, and confirms Orlin's conjecture for certain graph classes, advancing understanding of equistability.

Contribution

It provides a complete characterization of equistarable bipartite graphs and explores their relation to the triangle condition and equistability conjectures.

Findings

A bipartite graph is equistarable iff every 2-matching extends to a perfect matching.

Orlin's conjecture holds for complements of line graphs of bipartite graphs.

The triangle condition implies general partitionability in certain graph classes.

Abstract

Recently, Milani\v{c} and Trotignon introduced the class of equistarable graphs as graphs without isolated vertices admitting positive weights on the edges such that a subset of edges is of total weight $1$ if and only if it forms a maximal star. Based on equistarable graphs, counterexamples to three conjectures on equistable graphs were constructed, in particular to Orlin's conjecture, which states that every equistable graph is a general partition graph. In this paper we characterize equistarable bipartite graphs. We show that a bipartite graph is equistarable if and only if every $2$-matching of the graph extends to a matching covering all vertices of degree at least $2$. As a consequence of this result, we obtain that Orlin's conjecture holds within the class of complements of line graphs of bipartite graphs. We also connect equistarable graphs to the triangle condition, a combinatorial condition known to be necessary (but in general not sufficient) for equistability. We show that the triangle condition implies general partitionability for complements of line graphs of forests, and construct an infinite family of triangle non-equistable graphs within the class of complements of line graphs of bipartite graphs.