The matching number of tree and bipartite degree sequences

TL;DR

This paper characterizes the possible matching numbers in trees and bipartite graphs with given degree sequences, providing formulas and inequalities that determine realizability and matching number ranges.

Contribution

It introduces closed formulas for trees and an inequality-based characterization for bipartite graphs, extending Gale-Ryser Theorem insights.

Findings

Possible matching numbers form intervals for both trees and bipartite graphs.

Derived closed formulas for matching numbers in trees with given degree sequences.

Established an inequality-based criterion for bipartite degree sequences similar to Gale-Ryser.

Abstract

We study the possible values of the matching number among all trees with a given degree sequence as well as all bipartite graphs with a given bipartite degree sequence. For tree degree sequences, we obtain closed formulas for the possible values. For bipartite degree sequences, we show the existence of realizations with a restricted structure, which allows to derive an analogue of the Gale-Ryser Theorem characterizing bipartite degree sequences. More precisely, we show that a bipartite degree sequence has a realization with a certain matching number if and only if a cubic number of inequalities similar to those in the Gale-Ryser Theorem are satisfied. For tree degree sequences as well as for bipartite degree sequences, the possible values of the matching number form intervals.