$λ$-matchability in cubic graphs

Abstract

A vertex $v$ of a 2-connected cubic graph $G$ is $λ$-matchable if $G$ has a spanning subgraph in which $v$ has degree three whereas every other vertex has degree one, and we let $λ(G)$ denote the number of such vertices. Clearly, $λ=0$ for bipartite graphs; ergo, we define $λ$-matchable pairs analogously, and we let $ρ(G)$ denote the number of such pairs. We improve the constant lower bounds on both $λ$ and $ρ$ established recently by Chen, Lu and Zhang [Discrete Math., 2025] using matching-theoretic parameters arising from the seminal work of Lovász [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 each of whose vertices is $λ$-matchable.