$\hat{I}»$-matchability in cubic graphs

Abstract

A vertex $v$ of a 2-connected cubic graph $G$ is $\hat{I}»$-matchable if $G$ has a spanning subgraph in which $v$ has degree three whereas every other vertex has degree one, and we let $\hat{I}»(G)$ denote the number of such vertices. Clearly, $\hat{I}»=0$ for bipartite graphs; ergo, we define $\hat{I}»$-matchable pairs analogously, and we let $\ddot{I}(G)$ denote the number of such pairs. We improve the constant lower bounds on both $\hat{I}»$ and $\ddot{I}$ 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 $\hat{I}»$-matchable.

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