Another Proof of the Generalized Tutte--Berge Formula for $f$-Bounded Subgraphs

TL;DR

This paper provides a new proof of the generalized Tutte--Berge formula for the maximum size of $f$-bounded subgraphs in multigraphs, extending classical matching results to more general vertex degree constraints.

Contribution

The paper offers a novel proof of the min-max relation for $f$-bounded subgraphs using Tutte's $f$-Factor Theorem, generalizing the classical Tutte--Berge formula.

Findings

New proof of the generalized Tutte--Berge formula

Extension of classical matching theory to $f$-bounded subgraphs

Reinforcement of Tutte's $f$-Factor Theorem applications

Abstract

Given a nonnegative integer weight $f(v)$ for each vertex $v$ in a multigraph $G$, an {\it $f$-bounded subgraph} of $G$ is a multigraph $H$ contained in $G$ such that $d_H(v)\le f(v)$ for all $v\in V(G)$. Using Tutte's $f$-Factor Theorem, we give a new proof of the min-max relation for the maximum size of an $f$-bounded subgraph of $G$. When $f(v)=1$ for all $v$, the formula reduces to the classical Tutte--Berge Formula for the maximum size of a matching.