Isolated toughness and path-factor uniform graphs (II)

Sizhong Zhou (), Zhiren Sun () and Qiuxiang Bian ()
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Sizhong Zhou: Jiangsu University of Science and Technology
Zhiren Sun: Nanjing Normal University
Qiuxiang Bian: Jiangsu University of Science and Technology

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 3, 689-696

Abstract: Abstract A spanning subgraph F of G is called a path-factor if each component of F is a path. A $$P_{\ge k}$$ P ≥ k -factor of G means a path-factor such that each component is a path with at least k vertices, where $$k\ge 2$$ k ≥ 2 is an integer. A graph G is called a $$P_{\ge k}$$ P ≥ k -factor covered graph if for each $$e\in E(G)$$ e ∈ E ( G ) , G has a $$P_{\ge k}$$ P ≥ k -factor covering e. A graph G is called a $$P_{\ge k}$$ P ≥ k -factor uniform graph if for any two different edges $$e_1,e_2\in E(G)$$ e 1 , e 2 ∈ E ( G ) , G has a $$P_{\ge k}$$ P ≥ k -factor covering $$e_1$$ e 1 and avoiding $$e_2$$ e 2 . In other word, a graph G is called a $$P_{\ge k}$$ P ≥ k -factor uniform graph if for any $$e\in E(G)$$ e ∈ E ( G ) , the graph $$G-e$$ G - e is a $$P_{\ge k}$$ P ≥ k -factor covered graph. In this article, we demonstrate that (i) an $$(r+3)$$ ( r + 3 ) -edge-connected graph G is a $$P_{\ge 2}$$ P ≥ 2 -factor uniform graph if its isolated toughness $$I(G)>\frac{r+3}{2r+3}$$ I ( G ) > r + 3 2 r + 3 , where r is a nonnegative integer; (ii) an $$(r+3)$$ ( r + 3 ) -edge-connected graph G is a $$P_{\ge 3}$$ P ≥ 3 -factor uniform graph if its isolated toughness $$I(G)>\frac{3r+6}{2r+3}$$ I ( G ) > 3 r + 6 2 r + 3 , where r is a nonnegative integer. Furthermore, we claim that these conditions on isolated toughness and edge-connectivity in our main results are best possible in some sense.

Keywords: Graph; Isolated toughness; Edge-connectivity; Path-factor; Path-factor uniform graph.; 05C70; 05C38 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00286-x

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