ISOLATED TOUGHNESS VARIANT AND FRACTIONAL 𝑘 -FACTOR

. Isolated toughness is a crucial parameter considered in network security which characterizes the vulnerability of the network from the perspective of graph topology. 𝐼 ′ ( 𝐺 ) is the unique variant of isolated toughness which was introduced in 2001. This work investigates the correlation of 𝐼 ′ ( 𝐺 ) and the existence of fractional factor. It is proved that a graph 𝐺 with 𝛿 ( 𝐺 ) ≥ 𝑘 admits fraction 𝑘 -factor if 𝐼 ′ ( 𝐺 ) > 2 𝑘 − 1, where 𝑘 ≥ 2 is an integer. A counterexample is presented to show the sharpness of 𝐼 ′ ( 𝐺 ) bound


Introduction
This work only considers simple and finite graphs.Let  be a graph with vertex set  () and edge set ().We denote   () and   () (simply by () and  ()) as the degree and the neighborhood of  ∈  (), respectively.For any  ⊆  (), [] denotes the subgraph of  induced by , and set  −  = [ () ∖ ].Set () = min ∈ () {()} as the minimum degree of .The notations and terminologies used but undefined in this paper can be found in Bondy and Mutry [1].
Inspired by the idea of toughness, Yang et al. [8] introduced the notion of isolated toughness which is formalized as follows: () = +∞ if  is a complete graph; otherwise, where ( − ) is the number of isolated vertices in  − .A variant of isolated toughness was introduced by Zhang and Liu [10] which is formulated as

Z. HE ET AL.
Due to the theoretical importance and significant application of such parameters in specific fields, the investigation of isolation toughness in the setting of extended fractional factors (e.g., fractional (,  )-factor, all fractional factors, component factor), and in the setting of fractional deleted graph and fractional critical graph has attracted the attention from scholars.Ma and Liu [7] confirmed that a graph  admits a fractional -factor if () ≥  and () ≥ .Gao and Wang [2] determined an () bound for fractional (, , )-critical graphs.Gao et al. [3] studied the () condition for a graph which admits the component factor when the given numbers of edges are missing.Gao et al. [4] considered the isolated toughness parameter in 5-dimensional space, and computed the expression of detailed space structures.Zhou et al. [14] investigated the relationship between isolated toughness and path factors.More results on this topic and other extensions can be referred to [9,[11][12][13].
However, these extant results almost focus on original isolated toughness (), and there are few advances on  ′ ().Early studies found that () and  ′ () have obvious differences in parameter characteristics, while it is observed that most of the previously confirmed results for () are still open when considering  ′ () variant.For instance, the sharp () bound for a graph with fractional -factor was completely solved in 2006, and unfortunately, the tight  ′ () condition for the existence of fractional -factor is open till now.This tragic situation motivates us to do further in-depth research on  ′ ().
In this paper, we study the correlation between  ′ () and fractional -factor.Our main result can be formalized in the following theorem.
Obviously, () ≥  is tight for the existence of fractional -factor in terms of its definition.The following example reveals the sharpness of  ′ () bound in Theorem 1.Consider  = (2  ) ∨  1 where " ∨ " means a vertex in  1 adjacent to all vertices in 2  .Thus, we infer Set  =  ( 1 ) and  =  (2  ).We verify which implies that  has no fractional -factor in view of Lemma 1.
To prove Theorem 1, the following lemma which characterises the necessary and sufficient condition of fractional -factor is required.Lemma 1. ( Liu and Zhang [5]) Let  ≥ 1 be an integer.Then  has a fractional -factor if and only if Obviously, for a given subset  of  (), the subset  in Lemma 1 can be equivalently stated by  = { ∈  () − | − () ≤  − 1}.It is worthy to emphasize that Lemma 1 has its equal statement as follows.
Lemma 2. (Liu and Zhang [5]) Let  ≥ 1 be an integer.Then  has a fractional -factor if and only if holds for any disjoint subsets ,  ⊆  ().
The following two lemmas illustrate the properties of independent sets and covering sets in the specific conditions, which play a key role in the proof of the main theorem.Lemma 3. (Liu and Zhang [6]) Let  be a graph and let  = [ ] such that () ≥ 1 and 1 ≤   () ≤  − 1 for every  ∈  () where  ⊆  () and  ≥ 2. Let  1 , . . .,  −1 be a partition of the vertices of  satisfying   () =  for each  ∈   where we allow some   to be empty.If each component of  has a vertex of degree at most  − 2 in , then  has a maximal independent set  and a covering set  =  () −  such that where The following lemma is obtained by slightly modifying the Lemma 2.2 in [6] according to its proving process.Lemma 4. (Liu and Zhang [6]) Let  be a graph and let  = [ ] such that   () =  − 1 for every  ∈  () and no component of  is isomorphic to   where  ⊆  () and  ≥ 2. Then there exists an independent set  and the covering set  =  () −  of  satisfying where  () = { ∈ ,   () =  − } for 1 ≤  ≤  and ∑︀  =1 | () | = ||.

Proof of main result
If  is complete, the result is directly yielded by means of () ≥ .In what follows, we always assume that  is not complete.Suppose that  satisfies the conditions of Theorem 1, but has no fractional -factor.By Lemma 2, there exist disjoint subsets  and  of  () satisfying We select  and  such that | | is minimum.Thus, we immediately get  ̸ = ∅, and  − () ≤  − 1 for any  ∈  .
Let  be the number of the components of  ′ = [ ] which are isomorphic to   and let  0 = { ∈  ( ′ )| − () = 0}.Let  be the subgraph inferred from  ′ −  0 by deleting those  components isomorphic to   .Let  ′ be a set of vertices that contains exactly  − 1 vertices in each component of where  1 is the union of components of  which satisfies that  − () =  − 1 for each vertex  ∈  ( 1 ) and  2 =  −  1 .By means of Lemma 4,  1 has a maximum independent set  1 and the covering set and where Using the definitions of  and  2 , we verify that each component of  2 has a vertex of degree at most  − 2 in  − .According to Lemma 3,  2 has a maximal independent set  2 and the covering set where and where  0 = | 0 |.When ( −  ) ≥ 2, using the definition of  ′ (), we have If a contradiction.Therefore, (7) always established.

Conclusion and discussion
In this contribution, we obtain the tight  ′ () bound for a graph to admit fractional -factor.Since isolation toughness plays a key role in network security and the fractional factor is a characterization of fractional flow in data transmission networks, we believe that the theoretical conclusion determined in our paper has certain guiding significance for the practical application of network engineering.Furthermore, Theorem 1 has potential to be generalized in other fractional factor settings, as well as fractional deleted graph and fractional critical graph frameworks.Therefore, we propose the following open problems (the explanation of these concepts can be found in the relevant literatures).