Issue |
RAIRO-Oper. Res.
Volume 58, Number 5, September-October 2024
|
|
---|---|---|
Page(s) | 3849 - 3858 | |
DOI | https://doi.org/10.1051/ro/2024170 | |
Published online | 24 September 2024 |
Component factors and degree sum conditions in graphs
1
School of mathematics and big data, Anhui University of Science and Technology, Huainan 232001, P.R. China
2
School of Mathematical Science & Institute of Mathematics, Nanjing Normal University, Nanjing 210023, P.R. China
3
Research Center of Nonlinear Science, School of Mathematical and Physical Sciences, Wuhan Textile University, Wuhan 430073, P.R. China
4
College of Science, Nanjing Forestry University, Nanjing, Jiangsu 210037, P.R. China
5
School of Science, Hainan University, Haikou 570228, P.R. China
* Corresponding author: cheny@wtu.edu.cn
Received:
16
March
2024
Accepted:
18
August
2024
For a set 𝒜 of connected graphs, an 𝒜-factor is a spanning subgraph of a graph, whose connected components are isomorphic to graphs from the set 𝒜. An 𝒜-factor is also referred as a component factor. A graph G is called an (𝒜, m)-factor deleted graph if for every E′ ⊆ E(G) with |E′| = m, G − E′ admits an 𝒜-factor. A graph G is called an (𝒜, l)-factor critical graph if for every V′ ⊆ V (G) with |V′| = l, G − V′ admits an 𝒜-factor. Let m, l and k be three positive integers with k ≥ 2, and write ℱ = {P2, C3, P5, 𝒯 (3)} and ℋ = {K1,1, K1,2, . . . , K1,k, 𝒯 (2k + 1)}, where 𝒯 (3) and 𝒯 (2k + 1) are two special families of trees. Inspired by finding a sufficient condition to check for the existence of path-factors with some special restraints, we focus on the sufficient conditions based on a graphic parameter called degree sum: σκ(G) = minX⊆V(G) {Σx∈XdG(x):X is an independent set of k vertices}.
In this article, we verify that: (i) an (l + 2)-connected graph G of order n is an (ℱ, l)-factor critical graph if σ3(G) ≥ 6n+9l/5 ; (ii) a (2m + 1)-connected graph G of order n is an (ℱ, m)-factor deleted graph if σm+2(G) ≥ 6/5n ; (iii) an (l + 2)-connected graph G of order n is an (ℋ, l)-factor critical graph if σ2k+1(G) ≥ 6n+(6k+3)l/2k+3 ; (iv) a (2m + 1)-connected graph G of order n is an (ℋ, m)-factor deleted graph if σm+2(G) ≥ 6n/2k+3 .
Mathematics Subject Classification: 05C70 / 05C38
Key words: Graph / degree sum / component factor / factor critical graph / factor deleted graph
© The authors. Published by EDP Sciences, ROADEF, SMAI 2024
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.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.