Toughness Condition for a Graph to Be a Fractional -Critical Deleted Graph
A graph is called a fractional -deleted graph if admits a fractional -factor for any . A graph is called a fractional -critical deleted graph if, after deleting any vertices from , the resulting graph is still a fractional -deleted graph. The toughness, as the parameter for measuring the vulnerability of communication networks, has received significant attention in computer science. In this paper, we present the relationship between toughness and fractional -critical deleted graphs. It is determined that is fractional -critical deleted if .
All graphs considered in this paper are finite, are loopless, and are without multiple edges. The notation and terminology used but undefined in this paper can be found in . Let be a graph with the vertex set and the edge set . For a vertex , we use and to denote the degree and the neighborhood of in , respectively. Let denote the minimum degree of . For any , the subgraph of induced by is denoted by .
The problem of fractional factor can be considered as a relaxation of the well-known cardinality matching problem. It has wide-ranging applications in areas such as scheduling, network design, and the combinatorial polyhedron. For instance, several large data packets are to be sent to various destinations through several channels in a communication network. The efficiency of this work can be improved if large data packets are to be partitioned into small parcels. The feasible assignment of data packets can be seen as a fractional flow problem and it becomes a fractional factor problem when the destinations and sources of a network are disjoint.
Suppose that and are two integer-valued functions on such that for all . A fractional -factor is a function that assigns to each edge of a graph a number in  so that for each vertex we have . If and for all , then a fractional -factor is a fractional -factor. Moreover, if for all , then a fractional -factor is just a fractional -factor. Throughout this paper, is an integer, and we will not reiterate it again.
A graph is called a fractional -critical graph if, after deleting any vertices from , the resulting graph still has a fractional -factor. A graph is called a fractional -deleted graph if, after deleting any edges, the resulting graph still has a -factor. Fractional deleted graph and fractional critical graph, as extensions of the concept of fractional factor, describe the existence of fractional factor in communication networks when certain channels or certain sites are damaged.
Gao  proposed a new concept to deal with the combination situation when some channels and some sites are unavailable in networks. A graph is called a fractional -critical deleted graph if, after deleting any vertices from , the resulting graph is still a fractional -deleted graph. In particular, the fractional -critical deleted graph is just fractional -critical deleted graph if .
Let We heavily depend on the following lemma to prove our main result, which determined a necessary and sufficient condition for a graph to be a fractional -critical deleted graph.
Lemma 1 (Gao ). Let be a graph and let be two nonnegative integer-valued functions defined on satisfying for all . Let be a nonnegative integer. Then is a fractional -critical deleted graph if and only if for any disjoint subsets and of with .
The notion of toughness was first introduced by Chvátal in  to measure the vulnerable of networks: if is complete graph, ; if is not complete, where is the number of connected components of .
Liu and Zhang  determined a necessary and sufficient condition for a graph to have a fractional -factor. For several characters on fractional -factor one can refer to Liu and Zhang [4, 5] for more details. Liu  investigated the necessary and sufficient condition for a graph to be a fractional -critical graph. For more recent results for fractional deleted graph and fractional critical graph one can refer to [7–12].
Some toughness conditions for a graph to have a fraction factor were given in [13, 14]. Liu et al.  studied the relationship between toughness and fractional -critical graphs and proved that is a fractional -critical graph if for with and . Zhou et al.  studied the toughness condition for fractional ()-deleted graph. It is determined that is a fractional ()-deleted graph if . Recently, in , Gao et al. derived a new bound for graphs to be fractional -critical. It is verified that is a fractional -critical graph if . This inspires us to think about the for fractional -critical deleted graphs. In this paper, we determine that such bound of toughness as above is sufficient for a graph to be a fractional -critical deleted graph. Our main result to be proved in next section can be stated as follows.
Theorem 2. Let be a graph and let be two integer-valued functions defined on satisfying with and for all , where , are positive integers. Let be a nonnegative integer. if is complete. If , then is a fractional -critical deleted graph.
Clearly, our result strengthened the previous conclusions, and it is sharp if and according to the sharpness example in Liu and Zhang . The proof strategy is similar to the one in Liu and Zhang , but we need to cope with the more detailed case now and hence new methods are necessary. Before proving Theorem 2, we would like to show some useful lemmas.
Lemma 3 (Chvátal ). If a graph is not complete, then .
Lemma 4 (Liu and Zhang ). Let be a graph and let such that and for every , where and . Let be a partition of the vertices of satisfying for each , where one allows some to be empty. If each component of has a vertex of degree at most in , then has a maximal independent set and a covering set such that where and for every .
The lemma below can be deduced from Lemma 2.2 in .
Lemma 5 (Liu and Zhang ). Let be a graph and let such that for every and no component of is isomorphic to , where and . Then there exist an independent set and the covering set of satisfying where , , and .
2. Proof of Theorem 2
If is complete, then is a fractional -critical deleted graph due to . In what follows, we assume that is not complete.
Suppose that satisfies the conditions of Theorem 2 but is not a fractional -critical graph. By Lemma 1 and , there exist subsets and of such that We choose subsets and such that is minimum. Obviously, we deduce and for any .
Let be the number of the components of which are isomorphic to and let . Let be the subgraph obtained from by deleting those components isomorphic to .
If , then by virtue of (6) we infer or If , then , which contradicts and . If , then . Since , we have , which contradicts .
Now we consider that . Let , where is the union of components of which satisfies that for every vertex and . In terms of Lemma 5, has a maximum independent set and the covering set such that where , , and . Let for . Each component of has a vertex of degree at most in by the definitions of and . According to Lemma 4, has a maximal independent set and the covering set such that where and for every . Set and . We infer where . Let . Then when , we have and it also holds when . In terms of (11) and (12), we get In view of , we obtain Combining with (13), we deduce Therefore, By virtue of (9), we have Using (10), (16), and (17), we get
The following proof splits into two cases by the value of .
Case 1 (). By , we have . Hence, (18) becomes And then, at least one of the following two cases must hold.
Subcase 1 (). There is at least one such that
If , then . By , we get and , which contradicts the definition of and the choice of (see Lemma 4 proof in  such that ).
If , then , which contradicts .
Subcase 2 ( ). If or , then by we have
From and , if , we deduce
Furthermore, due to . Let
From , we infer
This is a contradiction.
If , we obtain Hence, a contradiction.
In conclusion, we have , , and (if , then and , a contradiction). Then the result follows from the main result in  which determined that is fractional 2-deleted graph if .
Case 2 (). In this case, (18) becomes
Subcase 1 (). In this subcase, (29) becomes Let
(i) If , then This implies , , , and . Thus, If , then and . This contradicts . Hence, .
Let and . Thus, and if . We obtain Letting , we infer By and , we get . By , we derive the contradiction.
(ii) If , then and the second largest value of is . In terms of the analysis of Lemma 4 in : for each connected component of , choose a vertex with the smallest degree and add it to . Hence, by the definition of , we confirm that is connected; each vertex in has degree in except that one vertex has degree in . This fact implies If , then and , which contradicts . Hence, and This reveals , which contradicts and .
Subcase 2 (). In this subcase, (29) becomes This implies Then, by , we get , , and . Now, we consider the following three subcases.
Subcase 2.1 (). In this subcase, we have . By analyzing the proof of Lemma 2.2 in : “for each vertex and , there exists a vertex such that ,” we obtain , Thus, This implies , a contradiction.
Subcase 2.2 (). In this subcase, . We can get a contradiction via a similar discussion as in Subcase 2.1.
Subcase 2.3). In this subcase, we have and . If , then . Thus, we infer a contradiction. Hence, . Let .
If there is a vertex such that only adjacent to one vertex in . Reset Then, we have By , we obtain This implies , a contradiction.
If each vertex in is adjacent to at least two vertices in , we get where . Due to , we deduce That is to say, , which contradicts and .
Subcase 3 ( and ). From what we have discussed in Subcase 1, we get . Then, we deduce This implies Thus, we have , , , and , by what we have discussed in Subcase 2. It is enough to discuss the situation of ; other two cases for and can be considered in a similar way.
Under the condition of , we get , , Since , we get Hence, This implies , a contradiction.
We complete the proof of the theorem.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
First the authors thank the reviewers for their constructive comments that helped improve the quality of this paper. They also would like to thank the anonymous referees for providing them with constructive comments and suggestions. This work was supported in part by Key Laboratory of Educational Informatization for Nationalities, Ministry of Education, the National Natural Science Foundation of China (60903131), Key Science and Technology Research Project of Education Ministry (210210), and the PHD initial funding of the first author.
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