Sufficient conditions for even factors in graphs

TL;DR

This work advances the study of even factors in graphs by providing tight sufficient conditions tied to both size and spectral radius for connected graphs with even order and minimum degree $\delta\ge2$. Building on Yan–Kano's Tutte-type framework, it identifies the extremal join graph $K_{\delta} \vee (K_{n-2\delta+1} \cup (\delta-1)K_1)$ as the threshold: if $e(G)\ge e(K_{\delta}\vee(K_{n-2\delta+1}\cup(\delta-1)K_1))$ or $\rho(G)\ge \rho(K_{\delta}\vee(K_{n-2\delta+1}\cup(\delta-1)K_1))$, then $G$ contains an even factor unless $G$ equals the extremal graph. The proofs leverage classic results on even factors and spectral radius, employing join-structure comparisons and equitable partitions to derive sharp inequalities that pinpoint the extremal configuration. The results yield precise, extremal-type criteria that enhance understanding of factor existence from both combinatorial and spectral viewpoints, with potential implications for related factor and spectral graph theory problems.

Abstract

Let $G$ be a graph. We denote by $e(G)$ and $ρ(G)$ the size and the spectral radius of $G$. A spanning subgraph $F$ of $G$ is called an even factor of $G$ if $d_F(v)\in\{2,4,6,\ldots\}$ for every $v\in V(G)$. Yan and Kano provided a sufficient condition using the number of odd components in $G-S$ for a graph $G$ of even order to contain an even factor, where $S$ is a vertex subset of $G$ [Z. Yan, M. Kano, Strong Tutte type conditions and factors of graphs, Discuss. Math. Graph Theory 40 (2020) 1057--1065]. In this paper, motivated by Yan and Kano's above result, we present some tight sufficient conditions to guarantee that a connected graph $G$ with the minimum degree $δ$ contains an even factor with respect to its size and spectral radius.