Cubic graphs with equal independence number and matching number

Abstract

Caro, Davila, and Pepper (arXiv:1909.09093) recently proved $\delta (G)\alpha (G)\le \Delta (G)\mu (G)$ 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 $\delta (G)=\Delta (G)=3$.