Isolated toughness and fractional (a,b,n)-critical graphs

A graph G is a fractional $ (a,b,n) $ (a,b,n)-critical graph if removing any n vertices from G, the resulting subgraph still admits a fractional $ [a,b] $ [a,b]-factor. In this paper, we determine the exact tight isolated toughness bound for fractional $ (a,b,n) $ (a,b,n)-critical graphs. To be specific, a graph G is fractional $ (a,b,n) $ (a,b,n)-critical if $ \delta (G)\ge a+n $ δ(G)≥a+n and $ I(G)>a-1+\frac {n+1}{n_{a,b}} $ I(G)>a−1+n+1na,b, where $ n_{a,b}\ge 2 $ na,b≥2 is an integer satisfies $ (n_{a,b}-1)a\le b\le n_{a,b}a-1 $ (na,b−1)a≤b≤na,ba−1. Furthermore, the sharpness of bounds is showcased by counterexamples. Our contribution improves a result from [W. Gao, W. Wang, and Y. Chen, Tight isolated toughness bound for fractional $ (k,n) $ (k,n)-critical graphs, Discrete Appl. Math. 322 (2022), 194–202] which established the tight isolated toughness bound for fractional $ (k,n) $ (k,n)-critical graphs.


Introduction
The graphs we discuss in this paper are simple and finite.Let G be a graph with vertex set V(G) and edge set E(G).We denote d G (v) and N G (v) (simply by d(v) and N(v)) as the degree and the neighbourhood of vertex v in G respectively, and δ(G) is the minimum degree of G which took the smallest value of d G (v) among all vertices.Let G[S] be the subgraph of G induced by S ⊆ V(G).Denote i(G) as the number of isolated vertices in G.The operator ∨ in G 1 ∨ G 2 means adding edges for all pairs of vertices between G 1 and G 2 .The notations and terminologies used but undefined in this paper can be referred to Bondy and Murty (2008).
Let a, b, k be positive integers with 1 ≤ a ≤ b, and h : E(G) → [0, 1] be an indicator function defined on the edge set.A fractional [a, b]-factor is a spanning subgraph consisting of edge set for any v ∈ V(G).A graph G admits a fractional [a, b]-factor means there is an indictor function h meeting (1).For n ∈ N, G is a fractional (a, b, n)-critical graph if removing any n vertices from G, the resulting subgraph still admits a fractional [a, b]-factor.In particular, if a = b = k, then fractional [a, b]-factor and fractional (a, b, n)-critical graph are degenerated to fractional k-factor and fractional (k, n)-critical graph, respectively, which are the original versions of fractional factor and fractional critical graph.As a salient graph parameter, isolated toughness is introduced by Yang et al. (2001) which is stated as if G is not a complete graph, and In recent two decades, with the advancement of graph theory technology and spread networks applicability, fractional factor has developed into a prominent research field in both graph theory and computer networks, and has achieved crucial theoretical results.Ma and Liu (2006) contended that a graph G admits a fractional k-factor if δ(G) ≥ k and I(G) ≥ k.Gao and Wang (2017) determined that G is a fractional (a, b, n)-critical graph if I(G) ≥ ab−2 a + n, where 2 ≤ a ≤ b and = b − a. Gao et al. (2022) proved the tight isolated toughness bound for a graph G to be fractional where k ≥ 2. Zhou (2021) and Zhou, Liu, Xu (2022) studied the graph parameter conditions of fractional critical covered graph which inquires fractional factors with given edges after removing fixed number of vertices.More results on this topic and other extensions can be referred to Chang et al. (2022), Chen et al. (2022), Zhou, Wu, Bian (2022), Zhou, Wu, Liu (2022) and Zhou, Wu, Xu (2022).
Although there are fruitful advances in the correlation of isolated toughness and the existence of fractional factor, the study of generalisation properties of fractional factors has been largely limited to the special setting, especially the tight I(G) bounds are open problems in most fractional critical graph settings.Motivated by the aforementioned facts, in this paper, we determine the sharp isolated toughness bound for a graph to be fractional (a, b, n)-critical, and our main conclusion is stated as follows which extends the previous result in Gao et al. (2022).
On the other hand, set S = V(K n+1 ) and Thus, G is not a fractional (a, b, n)-critical graph by means of Lemma 2.2.
By setting a = b = k in Theorem 1.1, n a,b = 2 and thus the following corollary is derived for fractional (k, n)-critical graphs.
Corollary 1.1 (Gao et al., 2022) The remainder of the paper is arranged as follows.Several useful lemmas and remarks are presented in the next section.Then, the detailed proof of Theorem 1.1 is presented.Finally, we discuss some interesting topics for future studies.

Preliminary
The necessary and sufficient condition of fractional (a, b, n)-critical graph is imperative to prove our main theorem which is determined by Liu (2010).
It is noteworthy that for a given subset S of V(G), T is determined by S which can be rewritten by We emphasise that Lemma 2.1 has its equal expression which is manifested below.The following lemma is obtained by slightly revising the Lemma 2.2 in Liu and Zhang (2008) in view of its proving process which presents the characteristic of independent sets and covering sets in specific settings.

Lemma 2.3 (Liu & Zhang, 2008): Let G be a graph and let H
and k ≥ 2 is an integer.Then there exists an independent set I and the covering set where The crux of our proofing strategy is to get the integer upper bound of |S| (or |U| instead), with the aim to eliminate the decimal part of |S| and obtain the desired conclusion.Another crucial trick in the proof process is to get the lower bound of the denominator of isolated toughness (|T 0 | + l + |I 1 | + |I 2 |, some terms may become zero in special cases).It is observed from the derivation in the next section that the lower bound of |T 0 | + l + |I 1 | + |I 2 | depends on n a,b , i.e. the correlation between a and b determines lower bound of denominator to maximise the I(G), and then we infer the contradiction.

Proof of theorem thm1.1
If G is complete, the result is directly yielded by means of δ(G) ≥ a + n.In what follows, we always assume that G is not complete.Suppose that G satisfies the conditions of Theorem 1.1, but is not a fractional (a, b, n)-critical graph.By Lemma 2.2, there exist disjoint subsets S and (2) We choose S and T such that |T| is minimum.Thus, T = ∅, and d G−S (x) ≤ a − 1 for any x ∈ T. Furthermore, due to δ(G) ≥ a + n, we obtain the following observation.
Observation 3.1: Let l be the number of the components of H = G[T] which are isomorphic to K a and let T 0 = {x ∈ V(H )|d G−S (x) = 0}.Let H be the subgraph inferred from H − T 0 by deleting those l components isomorphic to K a .Let S be a set of vertices that contains exactly a−1 vertices in each component of K a in H .
If |V(H)| = 0, then from (2) and Observation 3.1, we yield n arrives at the maximum value when |T 0 | + l reaches its lower bound n a,b .Hence, and where Using the definition of H and H 2 , we verify that each component of H 2 has a vertex of degree at most a−2 in G−S.Clearly, if H 2 = ∅, then a ≥ 3, and I 2 can be selected by Algorithm 1 (Gao et al., 2022): Choose a vertex v ∈ V(H 2 ) with the minimum degree in G − S (calculate the degree in the original subgraph G − S in each iteration).If all current vertices have degree a − 1 in original G − S, then select a vertex with minimum degree in the current subgraph. 3: Moreover, we have the following observation due to δ(G) ≥ a + n and at least one vertex in I 2 has degree at most a−2 in G−S.
The following two claims confirm that H 1 or H 2 can not exist independently.
Claim 3.1: We partition I 1 into two subsets.
In view of Observation 3.1, we deduce and thus . Hence, which contradicts to the hypothesis of I(G).

Claim 3.2:
It is clear to see that to acquire the extremum isolated toughness, we maximise |U| with the given number of |I 2 |, and thus only one vertex in I 2 has degree a−2 in G−S and others have degree a−1 in G−S.To bound the value of |T 0 | + l + |I 2 |, it is imperative to re-compute the upper bound of |S| as follows: Using Observation 3.2, we infer In terms of the definition of isolated toughness, we verify and n a,b ≥ 2.
From Claim 3.1 and Claim 3.2, we check Similar to the discussion in Claim 3.2, to maximise |U|, only one vertex in I 2 has degree a−2 in G−S and others have degree a−1 in G−S.To bound the value of By means of Observation 3.2, we yield

and hence
Using the definition of I(G), we obtain and n a,b ≥ 2. Case 2. |T 0 | + l = 0. Akin to Case 1, we verify that H 1 or H 2 can not exist independently.

Claim 3.3:
, and using the deduction in Claim 3.1, we get . Hence, is a non-negative term and it arrives at the maximum value when |I 1 | reaches its lower bound n a,b .We infer which contradicts to the hypothesis of isolated toughness.
According to Observation 3.2, we yield is a non-negative term and it arrives at the maximum value when |I 2 | reaches its lower bound 2n a,b − 2. We infer and n a,b ≥ 2.
It verifies from Claim 3.3 and Claim 3.4 that In terms of the similar deduction to Case 1, we confirm that |I 1 | + |I 2 | ≥ 2n a,b − 2 by Observation 3.2 and re-computing the supper bound of |S|.Therefore, we have and n a,b ≥ 2. Therefore, we complete the proof of Theorem 1.1.

Open problems
In this paper, we determine the sharp isolated toughness bound for a graph to be fractional (a, b, n)-critical.Since fractional [a, b]-factor is a specific case of the more general fractional factors.n the stage of computer network topology design, it needs to make a balance between network security and network construction cost.The denser the network, the greater the isolated toughness, the stronger the corresponding network's ability to resist attacks, and the stronger the network.However, such a high-density network requires high construction costs, which greatly increases the network construction and maintenance costs.The tight bound of the isolated toughness parameter of the fractional critical graph provides engineers an important reference index.The network constructed according to the tight bound can theoretically guarantee that data transmission can still be maintained when the network suffers certain damages, and at the same time ensure the minimum cost of network construction, thus cut the budget.
We raise the following problems for future studies (the concepts of all fractional factor and fractional (g, f )-factor can be referred to the other related papers, and we skip the detailed explanation here).

Theorem 1. 1 :
Let G be a graph, and a, b, n be positive integers with 2 ≤ a ≤ b and (n a,b − 1)a ≤ b ≤ n a,b a − 1, where n a,b ≥ 2 is an integer.If δ(G) ≥ a + n and I(G) > a − 1 + n+1 n a,b , then G is a fractional (a, b, n)-critical graph.To be obvious, δ(G) ≥ a + n is tight for a graph G to be fractional (a, b, n)-critical in terms of the definition of fractional [a, b]-factor.The sharpness of I(G) in Theorem 1.1 is showcased in the following instance.Consider G = K n+1 ∨ (n a,b K a ) where a, b, n are positive integers with b ≥ a ≥ 2, and n a,b ≥ 2 is an integer which is determined by the correlation b = (n a,b − 1)a + c with c ∈ {0, 1, . . ., a − 1}.We directly get Let a, b and n be positive integers such that a ≤ b.Then a graph G is fractional (a, b, n)-critical if and only if b|S| − a|T| + x∈T d G−S (x) ≥ bn holds for any S ⊆ V(G) with |S| ≥ n, where T Let a, b and n be positive integers such that a ≤ b.Then a graph G is fractional (a, b, n)-critical if and only if b|S| − a|T| + x∈T d G−S (x) ≥ bn holds for any disjoint subsets S, T ⊆ V(G) with |S| ≥ n.

Problem 4. 1 :
What is the tight isolated toughness bound for all fractional (a, b, n)-critical graphs?Problem 4.2: What is the tight isolated toughness bound for fractional (g, f , n)-critical graphs?