TOUGHNESS AND ISOLATED TOUGHNESS CONDITIONS FOR PATH-FACTOR CRITICAL COVERED GRAPHS

. 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 𝜏 ( 𝐺 ) > 𝑛 +23 ; (ii) 𝐺 is a ( 𝑃 ≥ 2 , 𝑛 )-factor critical covered graph if its isolated toughness 𝐼 ( 𝐺 ) > 𝑛 +12 ; (iii) 𝐺 is a ( 𝑃 ≥ 3 , 𝑛 )-factor critical covered graph if 𝜏 ( 𝐺 ) > 𝑛 +23 and | 𝑉 ( 𝐺 ) | ≥ 𝑛 + 3; (iv) 𝐺 is a ( 𝑃 ≥ 3 , 𝑛 )-factor critical covered graph if 𝐼 ( 𝐺 ) > 𝑛 +32 and | 𝑉 ( 𝐺 ) | ≥ 𝑛 + 3. Furthermore, we claim that these conditions are best possible in some sense.


Introduction
All graphs considered here are finite and simple.We refer to [5] for the notation and terminologies not defined here.Let  be a graph with vertex set  () and edge set ().Given a vertex  ∈  (), let   () denote the degree of , that the number of edges incident to  in .If   () = 0 for some vertex  in , then  is said to be an isolated vertex in .Let () be the number of isolated vertices of .For any subset  ⊆  (), let [] denote the subgraph of  induced by , and  −  := [ () ∖ ] is the resulting graph after deleting the vertices of  from .The number of connected components of a graph  is denoted by ().We write () for the vertex connectivity of .
Next, we introduce two parameters for a graph, namely, the toughness and the isolated toughness.The toughness of  was first introduced by Chvátal [6] as The isolated toughness of  was defined by Yang et al. [20] as if  is not complete; otherwise, () = +∞.A subgraph  of  is called a spanning subgraph of  if  () =  () and () ⊆ ().For a family of connected graphs ℱ, a spanning subgraph  of a graph  is called an ℱ-factor of  if each component of  is isomorphic to some graph in ℱ.A spanning subgraph  of a graph  is called a  ≥ -factor of  if every component of  is isomorphic to a path of order at least , where  ≥ 2 is an integer.For example, a  ≥3 -factor means a graph factor in which every component is a path of order at least three.
Akiyama et al. [2] provided a criterion for a graph having a  ≥2 -factor as follows.
In order to characterize a graph possessing a  ≥3 -factor, Kaneko [11] put forward the concept of a sun as follows.A graph  is called factor-critical if  −  has a 1-factor for each  ∈  ().Let  be a factorcritical graph and  () = { 1 ,  2 , . . .,   }.By adding new vertices { 1 ,  2 , . . .,   } together with new edges {    : 1 ≤  ≤ } to , the resulting graph is called a sun.Note that, according to Kaneko [11], we regard  1 and  2 also as a sun, respectively.Usually, the suns other than  1 and  2 are also called big suns.It is called a sun component of  if the component of  is isomorphic to a sun.We denote by sun() the number of sun components in .
Zhang and Zhou [22] first defined a graph  to be  ≥ -factor covered if  admits a  ≥ -factor containing  for any  ∈ ().In the same paper, they obtained a characterization for  ≥2 -factor covered graphs and  ≥3 -factor covered graphs, respectively.Theorem 1.3 (Zhang and Zhou [22]).Let  be a connected graph.Then  is a  ≥2 -factor covered graph if and only if ( − ) ≤ 2|| −  1 () for all  ⊆  (), where  1 () is defined by In this paper, we study ( ≥ , )-factor critical covered graphs and get some sufficient conditions for graphs to be ( ≥ , )-factor critical covered graphs depending on toughness and isolated toughness, which are given in Sections 2 and 3.

(𝑃 ≥2 , 𝑛)-factor critical covered graphs
In this section, using toughness and isolated toughness, we obtain two sufficient conditions for the existence of ( ≥2 , )-factor critical covered graphs.
Proof.If  is a complete graph, then it is easily seen that  is a ( ≥2 , )-factor critical covered graph by () ≥  + 1. Next, we consider that  is a non-complete graph.
For any To justify Theorem 2.1, it suffices to verify that  is  ≥2 -factor covered.Next, we assume that  is not  ≥2 -factor covered.Then by Theorem 1.3, there exists a subset  ⊆  () such that (1) Proof.On the contrary, we assume that  = ∅.Then  1 () = 0.It follows from (1) that It follows from (3) that  is an isolated vertex, which contradicts that | ()| ≥ 2. Hence, Claim 2.1 is verified.
Next, we will distinguish two cases below to complete the proof of Theorem 2.1.
Proof.If  is a complete graph, then it is easily seen that  is a ( ≥2 , )-factor critical covered graph by () ≥  + 1. Next, we consider that  is a non-complete graph.
Proof.If  is a complete graph, then it is easily seen that  is a ( ≥3 , )-factor critical covered graph by () ≥  + 1. Next, we consider that  is a non-complete graph.
For any  ′ ⊆  () with | ′ | = , we write  =  −  ′ .Clearly,  is connected.To justify Theorem 3.1, it suffices to verify that  is  ≥3 -factor covered.Next, we assume that  is not  ≥3 -factor covered.Then by Theorem 1.4, there exists a subset  ⊆  () such that Proof.On the contrary, we assume that  = ∅.Then  2 () = 0.It follows from ( 14) that Note that  is connected since It follows from (16) Using the definition of  (), we obtain which contradicts that  () > +2 3 in Theorem 3.1.Hence, Claim 3.1 is verified.
Next, we will distinguish two cases below to completes the proof of Theorem 3.1.