$λ$-matchability in cubic graphs

TL;DR

This work advances the theory of $\lambda$-matchability and $\lambda$-pairs in cubic graphs by leveraging parity arguments, Tutte's $1$-factor framework, and Lovász-inspired decomposition techniques. It yields constant lower bounds on $\lambda$ and $\rho$, with precise classifications of the tight examples via brick/brace decompositions, tight cuts, and splicing; it also provides a full characterization of $2$-connected cubic graphs with $\lambda=n$ through recursive constructions. The results extend from 3-connected to all 2-connected cubic graphs using a $2$-cut decomposition, and establish robust structure theorems that connect matchability to barrier fragments and contraction graphs. Overall, the paper deepens the structural understanding of spanning subgraphs with prescribed vertex degrees in cubic graphs and resolves the $\lambda=n$ characterisation, with clear implications for related matching-theoretic parameters $\B(G)$ and $\rho(G)$.

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 that satisfy $λ=n$.

Theorems & Definitions (74)

• proposition 1
• lemma 1
• proposition 2
• lemma 2
• proof
• proposition 3
• proposition 4
• proposition 5
• lemma 3
• proof