Forbidden Restrictions and the Existence of -Factor and -Factor

Abstract

The existence of factor and fractional factor in network graph in various settings has raised much attention from both mathematicians and computer scientists. It implies the availability of data transmission and network segmentation in certain special settings. In our paper, we consider -factor and -factor which are two special cases of general -factor. Specifically, we study the existence of these two kinds of path factor when some subgraphs are forbidden, and several conclusions on the factor-deleted graph, factor critical-covered graph, and factor uniform graph are given with regards to network parameters. Furthermore, we show that these bounds are best in some sense.

1. Introduction

All graphs considered in this work are finite simple graphs. Let be a graph, be the neighborhood of vertex , and . Let be the number of connected components in and . For the commonly used notations and terminologies, please refer to book [1] by Bondy and Mutry.

Let and be a path with at least vertices. A -factor is a spanning subgraph of such that each component is isomorphic to . A graph is a -factor-deleted graph if removing any edges from , the resting subgraph still admits -factor. For -factor, Akiyama et al. [2] demonstrated the following characteristic for its existence.

Lemma 1. A graph permits a -factor if and only if established for arbitrary vertex subset of .

More recent results on graph factors in various settings can be referred to Gao et al. [3, 4], Wang and Zhang and Zhou et al. [5–10], and Zhu et al. [11, 12].

A graph is factor-critical if deleting any vertex , the resulting subgraph has a perfect matching. A graph is called a sun if it is isomorphic to , , or the corona of a factor-critical graph, and the last class of sun is a big sun. Let be the number of sun components of . Kaneko [13] and Kano et al. [14] revealed that sun components can describe the existence of -factor, i.e, a graph admits a -factor if and only if for any vertex subset of .

Zhang and Zhou [15] introduced the concept of -factor-covered graph, i.e., a graph is -factor covered if for any edge , there is a -factor containing . Moreover, they obtained the following two conclusions for -factor-covered graph when or 3.

Lemma 2 (Zhang and Zhou [15]). A connected graph is a -factor-covered graph if and only if for any vertex subset of , where

Lemma 3 (Zhang and Zhou [15]). Assume as a connected graph. Then, is a -factor-covered graph if and only if for any , where

The concept of factor-covered graph can be further extended to factor-critical-covered graph. A graph is -factor-critical covered if deleted any vertices from , and the resting subgraph is still a -factor-covered graph.

In computer data communication networks, there are three main indices to test the robustness and vulnerability of networks, and also, there are some variables of these parameters. (i)Chvátal [16] firstly introduced toughness where if is complete; otherwise

Enomoto et al. [17] introduced a variant of toughness by revising the denominator to and denoted it by . That is to say, if is a complete graph; and for noncomplete graph. (ii)Isolated toughness was introduced by Yang et al. [18] as follows: if is a complete graph, then ; elsewise

Similar to , Zhang and Liu [19] introduced a variant of isolated toughness by revising the denominator to , denoted by : for a complete graph , and for others. (iii)Binding number is defined by Woodall [20] which is formulated by

The main contributions of this article are three folded: (1) the relationships between -factor-deleted graph and the above three parameters are studied; (2) toughness conditions for -factor-critical covered and -factor-critical covered graph are given; (3) toughness bounds for a graph to be -factor uniform graph and -factor uniform graph are determined. The main conclusions and detailed proofs are manifested in the next section, and then, in the third section, we present the sharpness of these bounds.

2. Main Results and Proofs

The purpose of this section is to present the main theorems and detailed proofs.

2.1. Bounds for -Factor-Deleted Graphs

Theorem 4. Let be a positive integer and be an -edge-connected graph. If (resp.) then is a -factor-deleted graph.

Proof. For a complete graph , the result follows from edge connectivity. Assume that is not complete, and clearly we have .

For arbitrary edge subset with edges, let , and we have and . We verify the theorem by means of proving that admits -factor. In contrast, we assume has no -factor, and hence, in view of Lemma 1, there is a subset of satisfying

If , then by (1) which contradicts to is -edge-connected and . Therefore, we infer and . Deleting one edge from , the number of its components adds most 1, thus .

We divide into three classes , , and .

If is a unique edge in which is a component in , then .

If and , one of end vertex of (say ) meets , then .

Otherwise, and at least one of its end vertices in , then .

We have and . Select one vertex in each edge in with larger degree in and denote by the set of these vertices. Thus, .

According to or accordingly we get .

For , we have

Let be a function with regard to . We have

Hence, is a monotonically increasing function and . We get which implies .

If , then and . If , then , a contradiction. Hence, is a connected graph, and there are at least 3 isolated vertices after removing 2 edges from . That is to say, which contradicts to is a 3-edge-connected graph.

For , we have

Let be a function with regard to . We obtain

Hence, is a monotonically increasing function and . We get which implies .

If , then and . If , then , a contradiction. Hence, is a connected graph, and there are at least three isolated vertices after removing two edges from . That is to say, which contradicts to that is a 3-edge-connected graph.

Hence, the proof of result is completed.

Theorem 5. Let be a positive integer and be an -edge-connected graph. If (resp. ), then, is a -factor-deleted graph.

Proof. For a complete graph , the result follows from edge connectivity. Assume that is not complete, and clearly, we have .

For arbitrary edge subset with edges, let , and we have and . We check the correctness of Theorem 5 via proving permits -factor. If not, we assume has no -factor, and hence ,using Lemma 1, there is a subset of satisfying (1).

If , then by (1) which contradicts to being -edge-connected and . Therefore, we infer and . Deleting one edge from , the number of its isolated vertices adds most 2; thus, .

We divide into three classes , , and as described in Theorem 4, and hence, and . Also, we use the same way to select vertex set , and thus, .

For , we have

Reset be a function with regard to . We acquire

Hence, is a monotonically increasing function and . Thus, we get a contradiction.

For , we have

Reset be a function with regard to . We acquire

Hence, is a monotonically increasing function and . Thus, we get a contradiction if .

Specially, if , then which implies . In this case, leads to which contradicts to being a 2-edge-connected graph.

Hence, the proof of this result is completed.

Theorem 6. Let be a positive integer and be an -edge-connected graph. If , then, is a -factor-deleted graph.

Proof. For a complete graph , the result follows from edge connectivity. Assume that is not complete, and clearly, .

Let for arbitrary edge subset with edges, and we have and . Assume that has no -factor, and hence, in view of Lemma 1, there is a subset of satisfying (1).

If , then, by (1) which contradicts to being -edge-connected and . Therefore, we infer and . Deleting one edge from , the number of its isolated components adds most 2, thus, .

Note that there are at least 3 isolated vertices after removing edges from . Also, since , we get , i.e., . Let be the vertex set of these isolated vertices in . If , we acquire a contradiction.

Now, we consider . If there is a vertex in meeting , then, set and since . We yield a contradiction.

If each vertex in has a degree at least 2 in , then, we can get the contradiction similar to what discussed above.

Hence, the proof of result is completed.

2.2. Toughness Conditions for -Factor-Critical Covered and -Factor-Critical Covered Graph

Theorem 7. Let and be a graph with . If , then, is a -factor critical covered graph.

Proof. If is complete, the result follows from . In what follows, we consider noncomplete graph.

For any with , set . To demonstrate is -factor critical covered, it is enough to prove is -factor covered. Otherwise, suppose is not -factor covered; then, according to Lemma 2, there is a vertex subset of such that

The following discussion is divided into three cases in terms of the value of .

Case 1. .
In this case, and by (2), which contradicts to .

Case 2. .
We consider the following two subcases.

Case 3. has no nontrivial component.

We infer and . By means of the definition of toughness, we deduce or a contradiction.

Case 4. has a nontrivial component.

We yield , , and . Using the definition of toughness, we have or a contradiction.

Case 5. .
We acquire and . In light of the definition of toughness, we obtain or a contradiction.
Therefore, the result follows.

Theorem 8. Let and be a graph with and . If , then, is a -factor critical covered graph.

Proof. If is a complete graph, then, the result follows from . We only consider noncomplete graph in what follows.
For any with vertices, let , and we aim to prove is -factor covered. On the contrary, is not a -factor covered graph, and then, by Lemma 3, there is a subset of meeting

The following discussion is divided into three cases by means of the value of .

Case 1. .
In this case, we summarize and by (3). Using and , we get . Since , we confirm that is a big sun. Let be the factor-critical graph of with and be a vertex in . Using the definition of toughness, we obtain or a contradiction.

Case 2. .
If there is a nonsun component of , we have , by (3), and . Directly using the definition of toughness, we yield or a contradiction.

If there is no nonsun component of , we get , by (3), and . In light of the definition of toughness, we infer or a contradiction.

Case 3. .
In this case, we acquire and in terms of (3). We verify or a contradiction.
Hence, Theorem 8 is verified.

Theorem 9. Let and be a graph with . If , then, is a -factor critical covered graph.

Proof. If is complete, we check the theorem using . Hence, we only consider noncomplete graph in the following contents.

For any with , set . To demonstrate that is -factor critical covered, it is enough to prove is -factor covered. Otherwise, suppose is not -factor covered; then, using Lemma 2, there is a vertex subset of satisfying (2).

The following discussion is divided into three cases in terms of the value of .

Case 1. .
In this case, we get contradiction as we discussed in Theorem 7.

Case 2. .
We consider the following two subcases.

Case 3. has no nontrivial component.
We infer and . By means of the definition of isolated toughness, we deduce or a contradiction.

Case 4. has nontrivial component.
We yield and . Using the definition of isolated toughness, we have or a contradiction.

Case 5. .
We acquire and . We can get the contradiction using the similar derivation to Theorem 7.
Therefore, we get the desired result.

Theorem 10. Let and be a graph with and . If (resp. ), then, is a -factor critical covered graph.

Proof. If is a complete graph, the result is hold from . We only discuss noncomplete graph in the following context.
For any with vertices, let , and we aim to prove is -factor covered. On the contrary, is not a -factor covered graph; then, using Lemma 3, there is a subset of satisfying (3).
The following discussion is divided into three cases according to how many elements in .

Case 1. .
In this case, similar to what’s discussed in Theorem 8, we have and , and is a big sun. Let be the factor-critical of with . Using the definition of , we obtain or a contradiction.

Case 2. .
We have . Suppose that there are ’s, ’s, and big sun components with in . Hence, by (3). We select one vertex from each and choose vertex set of factor-critical subgraph of every big sun and then denote by the vertex set of all these selected vertices. We infer and . In terms of the definition of isolated toughness, we yield It implies a contradiction.

For , we have

It implies a contradiction.

Case 3. .
In this case, we acquire and in terms of (3). Let be vertex subset defined as Case 2. We verify that is, It is implies that since , a contradiction.

For , we confirm which means,

It implies that , a contradiction.

Hence, Theorem 10 is verified.

Note that in Theorem 10. From Zhou et al. [21], we know that is a -factor covered graph if , and is tight.

2.3. Toughness Conditions for Factor Uniform Graph

A graph is a -factor uniform graph if for any two edges and , admits a -factor including and excluding . Zhou and Sun [?] studied the binding number condition for -factor uniform graph and -factor uniform graph. In this section, we research on other two parameters: toughness and isolated toughness. The idea to prove the following results is based on the observation that is -factor uniform if is -covered for any .

Theorem 11. Let be a 2-edge-connected graph. If , then, is a -factor uniform graph.

Proof. For any , is connected since is 2-edge-connected graph. To confirm Theorem 11, we need to verify that is -factor covered. If not, we assume that is not -factor covered. Using Lemma 2, there is a vertex subset of satisfying Furthermore, we have .
We consider three cases according to the value of .

Case 1. If .
We obtain which contradicts .

Case 2. If .
Then, and . If , then or a contradiction.
If , then, and . We infer or a contradiction.
If , then, is a component in and . If there is another component in except , then, , and we get the contradiction similar to the derivation above. If , then, since is 2-edge-connected graph. Special for , we yield , which is a -factor covered graph. Hence, satisfies the condition of theorem which is a -factor uniform graph.

Case 3. If .
Then, , and .
Notice that if , then, and . If , then, . Combining the above two cases, we have .
If , then or It implies , a contradiction.
If , then, using the fact that , we confirm that and are components in , , and . We acquire or Again, in both situation we get , which leads to a contradiction.

Theorem 12. Let be a 2-edge-connected graph. If , then, is a -factor uniform graph.

Proof. For any , is connected, and we only need to prove that is -factor covered. On the contrary, is not -factor covered, and we can find a subset of such that The following discussion is divided into three cases according to the value of .

Case 1. .
Then, and by (67). It implies , and is a big sun with at least six vertices. Moreover, is a graph constructed by adding an edge in a big sun. Let be the factor-critical of and . We have or a contradiction.

Case 2. .
Then, and by (67). If , then or a contradiction. If , then, , and it produces two sun components after deleting from . If isomorphic to , then, which is a -factor uniform graph. Otherwise, , and there are at least two vertices having degree 1 in . Let such that . We acquire or , a contradiction.

Case 3. .
In this case, , by (67), , and . If or , we deduce or a contradiction.

If , then, edge belongs to a nonsun component , while removing will produce two sun components. It means at least one of and is a cut vertex of component , and without loss of generality, we set as a cut vertex in . Hence, we get or a contradiction.

Thus, the proof of Theorem 12 is completed.

Theorem 13. Let be a 2-edge-connected graph. If , then, is a -factor uniform graph.

Proof. Clearly, we have . For any , is connected since is a 2-edge-connected graph. Similar as Theorem 11, we only need to verify that is -factor covered. In contrast, suppose that is not -factor covered. In terms of Lemma 2, there is a vertex subset of that meets (58). Furthermore, .
We consider three cases in view of the value of .

Case 1. .
We get which contradicts to .

Case 2. .
Then, and . If , then or a contradiction.
If , then, and assume . We infer or a contradiction.

If , then, is a component in and . If there is another component in except , then denote this component by . Select such that has a minimum degree in among all vertices in . Hence, and or a contradiction. If , then, becomes . As discussed in Theorem 11, meets the condition of Theorem 13 that is a -factor uniform graph.

Case 3. .
Then, , and . We consider the following subcases in light of the value of .

Case 4. .
If , then , or a contradiction.
If , then, , a contradiction. For , if , then a contradiction. If , we can easily check that , a contradiction.

Case 5. .
Since , we confirm that and are components in , and . Using , we acquire or a contradiction.
Thus, we confirm that Theorem 13 is established.

Theorem 14. Let be a 2-edge-connected graph. If , then, is a -factor uniform graph.

Proof. For any , is connected, and we only need to prove that is -factor covered. On the contrary, is not -factor covered. Then, there exists a subset of satisfying (67).

Let be the number of components, components, and big sun components in , respectively. Let be big sun components in with . Choosing one vertex from each component in and let be the set of these vertices. Set as the factor-critical subgraph of and . We have , and . The following discussion is divided into three cases according to the value of .

Case 1. .
Then, and by (67). It implies , is a big sun with at least six vertices, and . Moreover, is a graph constructed by adding an edge in a big sun. Let be the factor-critical of . We obtain or a contradiction.

Case 2. .
In this case, , by (67), and since and is 2-edge-connected.
Case 3. .

We get and (if , then , isomorphic to which contradicts to ).

If , using the definition of isolated toughness, we have which implies , a contradiction. For , we yield which implies , contradicting to .

If , then, and or which implies which contradicts to .

If , then, and contradicting to .

Case 4. .

In this case, since .

Claim 1. If is one of components in , then, .

Proof. Suppose is a component in and is exactly a deleted edge, set . If is isomorphic to , then, is isomorphic to which is clearly a -factor uniform graph.

If there is a component in , then, and or a contradiction.

If there is another component or big sun component in (say ), then, there is a vertex in such that and assume . We have and or a contradiction.

If there exists a nonsun component in (say ), the,n we select with its degree in as small as possible. We infer or a contradiction.

Hence, the claim is hold.

From Claim 1, we see that there is a nonsun component in with (and hence, ), delete edge from , and then, it produces a new sun component in . Thus, there is a vertex in with , and set . Note that , if is a component in , then, we yield or a contradiction. If or a big sun is a component in (denote this sun component by ), then, there is a vertex in with , and set . We acquire or a contradiction.

Case 3. .
In this case, there is a nonsun component in , and it produces two sun components after deleting from . Thus, there are at least two vertices such that . Set and note that and are allowed to be the same vertex (if ). If , then, , , and or a contradiction. Otherwise, , and or a contradiction.

Case 4. .
In this case, , by (67), and . We have , Then, the rest proof process is consistent with the part of Theorem 4 and Theorem 5 in Gao et al. [22], and we will not repeat here.
Hence, the proof of Theorem 14 is finished.

3. Sharpness

In this section, we present some counterexamples to verify that the bounds of parameters in theorems in the second section are tight.

3.1. Sharpness of Theorem 4-Theorem 6

We manifest that (1) and or in Theorem 4 cannot change to and (or ); (2) and or in Theorem 5 cannot change to and (or ); (3) and in Theorem 6 cannot change to and .

Let . Taking one vertex from each and denote by the set of these vertices, we have

Set and . Then, , and by setting , we have

Thus, has no -factor, and accordingly, is not a -factor-deleted graph.

3.2. Sharpness of Theorem 7 and Theorem 8

We show that the toughness bounds in Theorem 7 and Theorem 8 are best. Consider , and we have , and . Set with , and let . Take in , then, we have ,

Hence, according to Lemma 2, is not -factor covered, and is not a -factor critical covered graph. Moreover, in terms of Lemma 3, is not -factor covered, and is not a -factor critical covered graph.

3.3. Sharpness of Theorem 9

We depict that the isolated toughness bounds in Theorem 9 for a graph to be -factor critical covered are best. Consider where is enough large, and we have , and . Set with , and let . Set as the first in , then, we have and

Hence, by means of Lemma 2, is not -factor covered, and is not a -factor critical covered graph.

3.4. Sharpness of Theorem 10

The isolated toughness conditions in Theorem 10 are tight. Consider where is connected but not a sun. Set with , , and in . Selecting one vertex from each in the part and denoting by the set of these two vertices, we confirm

On the other hand, since is a nonsun component of and

In view of Lemma 3, is not -factor covered, and is not a -factor critical covered graph.

3.5. Sharpness of Theorem 11

The toughness bounds in Theorem 11 are tight. Consider which is 2-edge-connected graph with and . Select and set . Let . We have and

Therefore, by means of Lemma 2, is not -factor covered, and is not a -factor uniform graph.

3.6. Sharpness of Theorem 12

The isolated toughness bounds in Theorem 12 are sharp. Consider which is a 2-edge-connected graph. We have and . Let , , and be the vertex set of first in . We infer and

Hence, in terms of Lemma 3, is not -factor covered, and is not a -factor uniform graph.

3.7. Sharpness of Theorem 13 and Theorem 14

To show the isolated toughness bounds in Theorem 13 and Theorem 14 that are sharp, we consider where is a large number. Select one vertex from and vertices from and denote by the vertex subset of these vertices. We have

On the other hand, let and . Let be the vertex set of first in , and then, we have ,

Therefore, by means of Lemma 2, is not -factor covered, and is not a -factor uniform graph. Also, in terms of Lemma 3, is not -factor covered, and is not a -factor uniform graph.

4. Open Problems

The restrictions in factor critical graphs can be further extended to more general ones. For instance, a graph is a -factor critical covered graph if removing any vertices from , the resting subgraph is still a -factor covered graph (that is, if for any with , has a -factor containing all the edges in , and then, is called a -factor covered graph). The biggest obstacle to solve these problems is lacking of necessary and sufficient condition for -factor covered graph. Hence, as the first step, we need to expand the results on -factor covered graph and -factor covered graph determined by Zhang and Zhou [15] to necessary and sufficient condition of -factor covered graph and -factor covered graph. These problems are worthy of deep study in the future.

Data Availability

This work is a pure theoretical contribution, and no data are contained in the paper.

Conflicts of Interest

All authors declare no conflict of interests in publishing this work.

Acknowledgments

This research was funded by Jiangsu Provincial Key Laboratory of Computer Network Technology, School of Cyber Science and Technology.