Some results on path-factor critical avoidable graphs

Abstract

A path factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices. We write P ≥k = {P i : i ≥ k}. Then a P ≥k-factor of G means a path factor in which every component admits at least k vertices, where k ≥ 2 is an integer. A graph G is called a P ≥k-factor avoidable graph if for any e ∈ E(G), G admits a P ≥k-factor excluding e. A graph G is called a (P ≥k , n)-factor critical avoidable graph if for any Q ⊆ V (G) with |Q| = n, G − Q is a P ≥k-factor avoidable graph. Let G be an (n + 2)-connected graph. In this paper, we demonstrate that (i) G is a (P ≥2 , n)-factor critical avoidable graph if tough(G) > n+2 4 ; (ii) G is a (P ≥3 , n)-factor critical avoidable graph if tough(G) > n+1 2 ; (iii) G is a (P ≥2 , n)-factor critical avoidable graph if I(G) > n+2 3 ; (iv) G is a (P ≥3 , n)factor critical avoidable graph if I(G) > n+3 2. Furthermore, we claim that these conditions are sharp.

International Comparison on the Application of Information Technology in High School Mathematics Textbooks

“…A Zhou [27] acquired some toughness or isolated toughness conditions for graphs to be (P ≥k , n)-factor critical avoidable graphs for k = 2, 3.…”

“…3 , where n ≥ 0 is an integer. Theorem 4 ( [27]). An (n + 2)-connected graph G is (P ≥3 , n)-factor critical avoidable if its toughness t(G) > n+1 2 , where n ≥ 0 is an integer.…”

Two sufficient conditions for component factors in graphs

“…Zhang, Yan and Kano [11] posed a sufficient condition for the existence of {K 1,j , K 2k : k ≤ j ≤ 2k − 1}-factors in graphs. Zhou [14] derived some results on the existence of component factors in graphs. For the relationships between Two Sufficient Conditions for Component Factors in Graphs 3 binding number and graph factors, we refer the reader to [3,7,9,10,12,13,[15][16][17].…”

Characterizing an odd [1, b]-factor on the distance signless Laplacian spectral radius

“…Liu and Li [4] characterized a graph or a bipartite graph with a 1-factor by virtue of its distance signless Laplacian spectral radius. Lots of researchers presented some sufficient conditions on various parameters to guarantee the existence of [1,2]-factors in graphs, such as the neighborhood condition [5], the degree conditions [6][7][8], the binding number [8,9], the independence number [10,11], the isolated toughness [12][13][14][15], and the sun toughness [16]. Zhou [17] derived some sufficient conditions for graphs to possess [1,2]-factors.…”