Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2022; 59(4): 1045-1067

Published online July 31, 2022 https://doi.org/10.4134/BKMS.b210613

Copyright © The Korean Mathematical Society.

Complete characterization of odd factors via the size, spectral radius or distance spectral radius of graphs

Shuchao Li, Shujing Miao

Central China Normal University; Central China Normal University

Abstract

Given a graph $G,$ a $\{1,3,\ldots,2n-1\}$-factor of $G$ is a spanning subgraph of $G$, in which each degree of vertices is one of $\{1,3,\ldots,2n-1\}$, where $n$ is a positive integer. In this paper, we first establish a lower bound on the size (resp.~the spectral radius) of $G$ to guarantee that $G$ contains a $\{1,3,\ldots,2n-1\}$-factor. Then we determine an upper bound on the distance spectral radius of $G$ to ensure that $G$ has a $\{1,3,\ldots,2n-1\}$-factor. Furthermore, we construct some extremal graphs to show all the bounds obtained in this contribution are best possible.

Keywords: Odd factor, size, spectral radius, distance spectral radius

MSC numbers: Primary 05C70, 05C50, 05C72

Supported by: This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 12171190, 11671164).