Issue |
RAIRO-Oper. Res.
Volume 57, Number 2, March-April 2023
|
|
---|---|---|
Page(s) | 847 - 856 | |
DOI | https://doi.org/10.1051/ro/2023039 | |
Published online | 28 April 2023 |
Toughness and isolated toughness conditions for path-factor critical covered graphs
College of Science, Nanjing Forestry University, Nanjing, Jiangsu 210037, P.R. China
* Corresponding author: daiguowei1990@163.com; daiguowei@njfu.edu.cn
Received:
17
December
2021
Accepted:
28
March
2023
Given a graph G and an integer k ≥ 2. A spanning subgraph H of G is called a P≥k-factor of G if every component of H is a path with at least k vertices. A graph G is said to be P≥k-factor covered if for any e ∈ E(G), G admits a P≥k-factor including e. A graph G is called a (P≥k, n)-factor critical covered graph if G – V′ is P≥k-factor covered for any V′ ⊆ V(G) with |V′| = n. In this paper, we study the toughness and isolated toughness conditions for (P≥k, n)-factor critical covered graphs, where k = 2, 3. Let G be a (n + 1)-connected graph. It is shown that (i) G is a (P≥2, n)-factor critical covered graph if its toughness ; (ii) G is a (P≥2, n)-factor critical covered graph if its isolated toughness
; (iii) G is a (P≥3, n)-factor critical covered graph if
and |V(G)| ≥ n + 3; (iv) G is a (P≥3, n)-factor critical covered graph if
and |V(G)| ≥ n + 3. Furthermore, we claim that these conditions are best possible in some sense.
Mathematics Subject Classification: 05C70 / 05C38
Key words: Graph / Path-factor / Toughness / Isolated toughness / (P≥2 / n)-factor critical covered graph / (P≥3 / n)-factor critical covered graph
© The authors. Published by EDP Sciences, ROADEF, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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