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TOUGHNESS, ISOLATED TOUGHNESS AND PATH FACTORS IN GRAPHS

Part of: Graph theory

Published online by Cambridge University Press:  03 December 2021

SIZHONG ZHOU Open the ORCID record for SIZHONG ZHOU [Opens in a new window] ,
JIANCHENG WU  and
YANG XU
SIZHONG ZHOU*
Affiliation:
School of Science, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212100, China
JIANCHENG WU
Affiliation:
School of Science, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212100, China e-mail: wujiancheng@just.edu.cn
YANG XU
Affiliation:
Department of Mathematics, Qingdao Agricultural University, Qingdao, Shandong 266109, China e-mail: xuyang_825@126.com
*
e-mail: zsz_cumt@163.com
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Abstract

A graph G is called a $(P_{\geq n},k)$-factor-critical covered graph if for any $Q\subseteq V(G)$ with $|Q|=k$ and any $e\in E(G-Q)$, $G-Q$ has a $P_{\geq n}$-factor covering e. We demonstrate that (i) a $(k+1)$-connected graph G with at least $k+3$ vertices is a $(P_{\geq 3},k)$-factor-critical covered graph if its toughness $t(G)>{(2+k)}/{3}$; (ii) a $(k+2)$-connected graph G is a $(P_{\geq 3},k)$-factor-critical covered graph if its isolated toughness $I(G)>{(5+k)}/{3}$. Furthermore, we show that the conditions on $t(G)$ and $I(G)$ are sharp.

Keywords

graphtoughnessisolated toughnessP≥3-factor(P≥3, k)-factor-critical covered graph

MSC classification

Primary: 05C70: Factorization, matching, partitioning, covering and packing
Secondary: 05C38: Paths and cycles

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 195 - 202
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported by the Six Talent Peaks Project in Jiangsu Province, China (Grant No. JY–022).

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