An upper bound on the distinguishing index of graphs with minimum degree at least two

An upper bound on the distinguishing index of graphs with minimum degree at least two

Saeid Alikhani 111Corresponding author    Samaneh Soltani
July 20, 2019
Abstract

The distinguishing index of a simple graph , denoted by , is the least number of labels in an edge labeling of not preserved by any non-trivial automorphism. It was conjectured by Pilśniak (2015) that for any 2-connected graph . We prove a more general result for the distinguishing index of graphs with minimum degree at least two from which the conjecture follows. Also we present graphs for which .

Department of Mathematics, Yazd University, 89195-741, Yazd, Iran

alikhani@yazd.ac.ir, s.soltani1979@gmail.com

Keywords: distinguishing index; edge colourings; bound

AMS Subj. Class.: 05C25, 05C15

1 Introduction

Let be a simple connected graph. We use the standard graph notation ([4]). In particular, denotes the automorphism group of . For simple connected graph , and , the neighborhood of a vertex is the set . The degree of a vertex in a graph , denoted by , is the number of edges of incident with . In particular, is the number of neighbours of in . We denote by and the minimum and maximum degrees of the vertices of . A graph is -regular if for all . The diameter of a graph is the greatest distance between two vertices of , and denoted by .

The distinguishing index of a graph is the least number such that has an edge labeling with labels that is preserved only by the identity automorphism of . The distinguishing edge labeling was first defined by Kalinowski and Piśniak [6] for graphs (was inspired by the well-known distinguishing number which was defined for general vertex labelings by Albertson and Collins [1]). The distinguishing index of some examples of graphs was exhibited in [6]. For instance, for every , and for , for . Also, for complete graphs , we have for , for . They showed that if is a connected graph of order and maximum degree , then , unless is or . It follows for connected graphs that if and only if and is a cycle of length at most five. The equality holds for all paths, for cycles of length at least 6, for , and for symmetric or bisymmetric trees. Also, Pilśniak showed that for all other connected graphs.

Theorem 1.1

[7] Let be a connected graph that is neither a symmetric nor an asymmetric tree. If the maximum degree of is at least 3, then unless is or .

Pilśniak put forward the following conjecture.

Conjecture 1.2

[7] If is a -connected graph, then .

In this paper, we prove the following theorem which proves the conjecture.

Theorem 1.3

Let be a connected graph of maximum degree . If the minimum degree , then .

For our purposes, we consider graphs with specific construction that are from dutch-windmill graphs. Because of this, in Section 2, we compute the distinguishing index of the dutch windmill graphs. In Section 3, we use the results to prove the main result. In the last section we present graphs for which .

2 Distinguishing index of dutch windmill graphs

To obtain the upper bound for the distinguishing index of connected graphs with minimum degree at least two, we characterize such graphs with minimum number of edges. For this characterization we need the concept of dutch windmill graphs. The dutch windmill graph is the graph obtained by taking , () copies of the cycle graph , () with a vertex in common (see Figure 1). If , then we call , a friendship graph. In the following theorem we compute the distinguishing number of dutch windmill graphs.

Figure 1: Examples of dutch windmill graphs.
Theorem 2.1

For every and , .

Proof. We consider two cases:

Case 1) If is odd. There is a natural number such that . We can consider a blade of as Figure 2.

Figure 2: The considered polygon (or a cycle of size ) in the proof of Theorem 2.1.

Let be the label of vertices of the th blade where . Suppose that is a labeling of the vertices of except its central vertex. In an -distinguishing labeling we must have:

  • There exists such that for all .

  • For we must have and .

There are possible -arrays of labels using labels satisfying (i) and (ii), and so .

Case 2) If is even. There is a natural number such that . We can consider a blade of as Figure 2. Let be the label of vertices of th blade where . Suppose that is a labeling of the vertices of except its central vertex. In an -distinguishing labeling we must have:

  • There exists such that for all .

  • For we must have

There are possible -arrays of labels using labels satisfying (i) and (ii) ( choices for and choices for ). Therefore .

The following theorem gives the distinguishing index of .

Theorem 2.2

For any and , .

Proof. Since the effect of every automorphism of on its non-central vertices is exactly the same as the effect of an automorphism of on its edges and vice versa, so if we consider the non-central vertices of as the edges of , then we have . Therefore the result follows from Theorem 2.1.

3 Proof of conjecture

In this section, we shall prove Conjecture 1.2. To do this, first we state some preliminaries. By the result obtained by Fisher and Isaak [3] and independently by Imrich, Jerebic and Klavžar [5] the distinguishing index of complete bipartite graphs is as follows:

Theorem 3.1

[3, 5] Let be integers such that and . Then

If then the distinguishing index is either or and can be computed recursively in time.

Corollary 3.2

[7] If , then .

Also we need the following result:

Theorem 3.3

[7] If is a graph of order such that has a Hamiltonian path, then .

Now, we state and prove the main theorem of this paper.

Theorem 3.4

Let be a connected graph with maximum degree . If then .

Proof. If , then the result follows from Theorem 1.1. So, we suppose that . Let be a vertex of with the maximum degree . By Theorem 2.2, we can label the edges of the dutch windmill graph attached to at vertex (a subgraph is attached to graph , if it has only one vertex in common with graph ) for which is the central point of the dutch windmill graph, with at most labels from label set , distinguishingly. If there exists triangle attached to at , then we label the two its incident edges to with and , and another edges of the triangle with label .

Let be the vertices of at distance one from , except the vertices of dutch windmill or triangle attached to graph at . We continue the labeling by the following steps:

Step 1) Since , so for and , we label the edges with label , and we do not use the label any more. With respect to the number of incident edges to with label 0, we conclude that the vertex is fixed under each automorphism of preserving the labeling. Also, since the dutch windmill or the triangle graph attached to at has been labeled distinguishingly, so the vertices of attached graph are fixed under each automorphism of preserving the labeling. Hence, every automorphism of preserving the labeling must map the set of vertices of at distance from to itself setwise, for any . We denote the set of vertices of at distance from , by for . If for any , , then has a Hamiltonian path, and since , so the order of is at least , and hence by Theorem 3.3. Thus we suppose that , for some .

Now we partition the vertices of to two sets and as follows:

The sets and are mapped to and , respectively, setwise, under each automorphism of preserving the labeling. For , we set . By this notation, we get that for , the set is mapped to setwise, under each automorphism of preserving the labeling. Let the sets and for are as follows:

It is clear that the sets and are mapped to and , respectively, setwise, under each automorphism of preserving the labeling. Since for any , we have , so we can label all incident edges to each element of with labels , such that for any two vertices of , say and , there exists a label , , such that the number of label for the incident edges to vertex is different from the number of label for the incident edges to vertex . Hence, it can be deduce that each vertex of is fixed under each automorphism of preserving the labeling, where . Thus every vertices of is fixed under each automorphism of preserving the labeling. In sequel, we want to label the edges incident to vertices of such that is fixed under each automorphism of preserving the labeling, pointwise. For this purpose, we partition the vertices of to the sets , () as follows:

Since the set , for any , is mapped to itself, it can be concluded that is mapped to itself, setwise, under each automorphism of preserving the labeling, for any and . Let . It is clear that . Let , and . We assign to the -ary of edges, a -ary of labels such that for every and , , there exists a label in their corresponding -arys of labels for which the number of label in the corresponding -arys of and is distinct. For constructing numbers of such -arys we need, distinct labels. Since for any , we have

so we need at most distinct labels from label set for constructing such -arys. For instance, let , and . By our method, we label the edge with label for where . Hence, the vertices of , for any , are fixed under each automorphism of preserving the labeling. Therefore, the vertices of for any , and so the vertices of are fixed under each automorphism of preserving the labeling. Now, we can get that all vertices of are fixed. If there exist unlabeled edges of with the two endpoints in , then we assign them an arbitrary label, say 1.

Step 2) Now we consider . We partition this set such that the vertices of with the same neighbours in , lie in a set. In other words, we can write , such that contains that elements of having the same neighbours in , for any . Since all vertices in are fixed, so the set is mapped to setwise, under each automorphism of preserving the labeling. Let , and we have

We consider two following cases:

Case 1) If for every and in , where , there exists a , , for which the label of edge is different from label of edge , then all vertices of in are fixed under each automorphism of preserving the labeling.

Case 2) If there exist and in , where , such that for every , , the label of edgeس and are the same, then we can make a labeling such that the vertices in have the same property as Case 1, and so are fixed under each automorphism of preserving the labeling, by using at least one of the following actions:

  • By commutating the coordinates of -ary of labels assigned to the incident edges to with an end point in .

  • By using a new -ary of labels, with labels , for incident edges to with an end point in , such that (by notations in Step 1) for every and , , there exists a label in their corresponding -arys of labels with different number of label in their coordinates, where .

  • By labeling the unlabeled edges of with the two end points in which are incident to the vertices in .

  • By labeling the unlabeled edges of which are incident to the vertices in , and another their endpoint is .

  • By labeling the unlabeled edges of with the two end points in for which the end points in are adjacent to some of vertices in .

Using at least one of above actions, it can be seen that every two vertices and in have the property as Case 1. Thus we conclude that all vertices in , for any , and so all vertices in , are fixed under each automorphism of preserving the labeling. If there exist unlabeled edges of with the two endpoints in , then we assign them an arbitrary label, say 1.

By continuing this method, in the next step we partition exactly by the same method as partition of to the sets ’s in Step 2, and so we can make a labeling such that is fixed pointwise, under each automorphism of preserving the labeling, for any .

For a -connected planar graph , the distinguishing index may attain . For example, consider the complete bipartite graph with , where is a positive integer . By Theorem 3.1, .

4 Graphs with

In this section, we present graphs with specific construction such that . To do this we state the following definition.

Definition 4.1

Let be a connected graph with . The graph is called a -minimally graph, if the minimum degree of each spanning subgraph of is less than .

It can be concluded from Definition 4.1 that if is an edge of a connected -minimally graph with end points and , then without loss of generality we can assume that and . In fact the distance between the two vertices of degree greater than is at least two.

The simplest connected -minimally graphs are cycles and complete bipartite graphs . Now, we explain more on the structure of a -minimally graph. Let to call a path in the graph a simple path, if all its internal vertices have degree two. Let be a connected -minimally graph.

  • If the degree of all vertices of is two, then is a cycle graph.

  • If there exist a vertex of with degree at least three. We consider two following cases:

    Case 1) If is the only vertex of with degree greater than two, then is a graph which is made by identifying the central points of some dutch windmill graphs where , and hence where is a set of indices. In this case we denote by (for instance, see Figure 3).

    Figure 3: Examples of .

    Case 2) If has other vertex of degree greater than two, then there exists at least a simple path between and of length greater than one. Since is a connected graph, so if there exists no such simple path, then there exists a vertex of degree at least three on each path between and . Hence we can obtain a vertex of with degree greater than two such that there exists at least a simple path between and of length greater than one (see Figure 4).

Figure 4: The state of vertices of degree greater than two in a connected -minimally graph.

Now we characterize graphs with .

Theorem 4.2

Let be a connected -minimally graph with maximum degree . If is not a cycle or a complete bipartite graph for some integer , then .

Proof. If , then is a cycle. It is known that the distinguishing index of cycle graph of order at least is two. Hence, we suppose that is not a cycle, so . Let be a vertex of of maximum degree . Suppose that are all vertices of which are of degree at least three such that there exists at least a simple path between and , for any (it is possible that ). Let there exist disjoint simple paths of length between and , for any and where is a non-negative integer and . We can label these simple paths of length with at most labels, by using numbers of -arys such that the coordinates of each -ary are in the set , for any and , and for every two paths of length , say and , there exists a label , , such that the number of label in -arys related to and are distinct. Let be a simple path between and for some , such that the label of edge of which is incident to , is different from the label of edge of which is incident to . We do not use of labeling of the simple path , for any other simple path (with the same length) between any two vertices of degree greater than two. Since is not a complete bipartite graph for some integer , so we can label these paths distinguishingly with at most labels. Now, we label the induced subgraph , for any vertex of degree greater than two, if there exists, with at most labels distinguishingly by Theorem 2.2, such that the distinguishing labeling of is nonisomorphic to the remaining distinguishing labeling of , where . Thus any automorphism of preserving this labeling should be fixed and all vertices of degree two on the simple paths between and for any . Since for any where , the vertices and are fixed, so all the simple paths between and , if there exist, are mapped to each other under each automorphism of preserving this labeling. Hence we can label all edges of these simple paths with at most labels, by assigning distinct ordered arys of labels of length of the simple paths between and such that all vertices of these paths are fixed under each automorphism of preserving this labeling.

For any , we consider , and suppose that are all vertices of with degree at least three such that there exists at least a simple path between and for any . Now we do the same method as labeling of simple paths between and , for all simple paths between and with at most labels. Also, we do the same method as labeling of simple paths between and , for all simple paths between and with at most labels, where . Note that we do not use labeling of for any simple path with the same length as between and . Thus the vertices and all vertices of the simple paths between them are fixed under each automorphism of preserving this labeling. After the finite number of steps we can obtain a distinguishing edge labeling of with at most labels.

5 Conclusion

We gave an upper bound for the distinguishing index of graphs with minimum degree at least two. This result proves a conjecture by Pilśniak (2015). We also studied graphs with . We think that the following conjecture is true, but until now all attempts to prove this failed. So, we end this paper by proposing the following conjecture.

Conjecture 5.1

Let be a connected graph with maximum and minimum degree and , respectively.

  • If is a -minimally graph with such that is not a complete bipartite or -regular graph, then .

  • If is a connected graph with , then .

References

  • [1] M.O. Albertson and K.L. Collins, Symmetry breaking in graphs, Electron. J. Combin. 3 (1996), #R18.
  • [2] S. Alikhani and S. Soltani, Distinguishing number and distinguishing index of certain graphs, Filomat, to appear. http://arxiv.org/abs/1602.03302.
  • [3] M. J. Fisher and G. Isaak, Distinguishing colorings of Cartesian products of complete graphs, Discrete Math. 308 (11) (2008), 2240-2246.
  • [4] R. Hammack, W. Imrich and S. Klav̌zar, Handbook of product graphs (second edition), Taylor & Francis group, 2011.
  • [5] W. Imrich, J. Jerebic and S. Klavžar, The distinguishing number of Cartesian products of complete graphs, European J. Combin. 29 (2008), 922-929.
  • [6] R. Kalinowski and M. Pilśniak, Distinguishing graphs by edge colourings, European J. Combin. 45 (2015), 124-131.
  • [7] M. Pilśniak, Nordhaus-Gaddum bounds for the distinguishing index, (2015) Available at www.ii.uj.edu.pl/preMD/.
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