$\lambda$-matchability in cubic graphs

TL;DR

This paper investigates $ ext{lambda}$-matchability in 2-connected cubic graphs, improving bounds on matchable vertices and pairs, characterizing extremal cases, and solving an open problem about graphs with maximum matchability.

Contribution

It improves lower bounds on $ ext{lambda}$ and $ ho$, characterizes tight examples, and fully characterizes 2-connected cubic graphs with $ ext{lambda} = n$, advancing understanding of matchability in cubic graphs.

Findings

Improved constant lower bounds on $ ext{lambda}$ and $ ho$

Characterization of all tight examples for these bounds

Complete characterization of graphs with $ ext{lambda} = n$

Abstract

A vertex $v$ of a 2-connected cubic graph $G$ is $\lambda$-matchable if $G$ has a spanning subgraph in which $v$ has degree three whereas every other vertex has degree one, and we let $\lambda(G)$ denote the number of such vertices. Clearly, $\lambda=0$ for bipartite graphs; ergo, we define $\lambda$-matchable pairs analogously, and we let $\rho(G)$ denote the number of such pairs. We improve the constant lower bounds on both $\lambda$ and $\rho$ established recently by Chen, Lu and Zhang [Discrete Math., 2025] using matching-theoretic parameters arising from the seminal work of Lov\'asz [J. Combin. Theory Ser. B, 1987], and we characterize all of the tight examples. We also solve the problem posed by Chen, Lu and Zhang: characterize 2-connected cubic graphs that satisfy $\lambda=n$.