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Spectral radius and size conditions for 1-tough balanced bipartite graphs to contain a k-factor with k≥ 3
Abstract
The largest eigenvalue of the adjacency matrix A(G) of a graph G is called the spectral radius ρ(G) of G. If S is a vertex cut of a connected graph G, then c(G−S) denotes the number of connected components of G−S. The toughness t(G) of a graph G is defined as t(G) = min{|S|/(c(G−S)): S ⊂ V(G), and c(G−S) > 1}. A graph G is t-tough if |S| ≥ t c(G − S) for every proper subset S ⊂ V(G) satisfying c(G − S) > 1. Combining the toughness and spectral conditions, we provide spectral radius and size conditions for the existence of k-factors in 1-tough balanced bipartite graphs. This settles a problem posed by Chen, Fan and Lin in [Discrete Appl. Math. {\bf 355} (2024), 223--231].
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BibTeX
@misc{ren2026spectral,
title = {{Spectral radius and size conditions for 1-tough balanced bipartite graphs to contain a $k$-factor with $k\ge 3$}},
author = {Jinxuan Ren and Jiayao Song and Ligong Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02615},
eprint = {2611.02615},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02615},
}
RIS
TY - UNPB
TI - Spectral radius and size conditions for 1-tough balanced bipartite graphs to contain a k-factor with k≥ 3
AU - Jinxuan Ren
AU - Jiayao Song
AU - Ligong Wang
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02615
UR - https://arxiv.org/abs/2611.02615
AB - The largest eigenvalue of the adjacency matrix A(G) of a graph G is called the spectral radius ρ(G) of G. If S is a vertex cut of a connected graph G, then c(G−S) denotes the number of connected components of G−S. The toughness t(G) of a graph G is defined as t(G) = min{|S|/(c(G−S)): S ⊂ V(G), and c(G−S) > 1}. A graph G is t-tough if |S| ≥ t c(G − S) for every proper subset S ⊂ V(G) satisfying c(G − S) > 1. Combining the toughness and spectral conditions, we provide spectral radius and size conditions for the existence of k-factors in 1-tough balanced bipartite graphs. This settles a problem posed by Chen, Fan and Lin in [Discrete Appl. Math. {\bf 355} (2024), 223--231].
ER -