Cubic graphs with equal independence number and matching number

TL;DR

This paper characterizes extremal cubic graphs where the independence number equals the matching number, building on a recent inequality relating degrees, independence, and matching numbers.

Contribution

It provides a complete characterization of extremal cubic graphs with equal independence and matching numbers, addressing open problems from prior research.

Findings

Characterization of extremal cubic graphs with equal independence and matching numbers

Resolution of open problems for graphs with degrees 3 and unequal degrees

Extension of inequality relating degrees, independence, and matching numbers

Abstract

Caro, Davila, and Pepper (arXiv:1909.09093) recently proved $\delta(G) \alpha(G)\leq \Delta(G) \mu(G)$ for every graph $G$ with minimum degree $\delta(G)$, maximum degree $\Delta(G)$, independence number $\alpha(G)$, and matching number $\mu(G)$. Answering some problems they posed, we characterize the extremal graphs for $\delta(G)<\Delta(G)$ as well as for $\delta(G)=\Delta(G)=3$.