The chromatic distinguishing index of certain graphs

The chromatic distinguishing index of certain graphs

Abstract

The distinguishing index of a graph , denoted by , is the least number of labels in an edge labeling of not preserved by any non-trivial automorphism. The distinguishing chromatic index of a graph is the least number such that has a proper edge labeling with labels that is preserved only by the identity automorphism of . In this paper we compute the distinguishing chromatic index for some specific graphs. Also we study the distinguishing chromatic index of corona product and join of two graphs.

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

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

Keywords: distinguishing index; edge colorings; chromatic index.

AMS Subj. Class.: 05C25, 05C15

1 Introduction

Let be a simple graph and denote the automorphism group of . For , the neighborhood of is the set . The degree of in a graph , denoted by , is the number of edges of incident with . In particular, is the number of neighbours of in . Also, the maximum degree of is denoted by .

A proper edge labeling of a nonempty graph (a graph with edges) is a function , where is a set of labels (colors), with the property that for every two adjacent edges and of . If the labels are chosen from a set of labels, then is called a proper -edge labeling of . The minimum positive integer for which has a proper -edge labeling is called the chromatic index of and is denoted by . As a result of Vizing’s theorem, the chromatic index of every nonempty graph is one of two numbers, namely or . A graph with is called a class one graph while a graph with is called a class two graph. For instance, it is proved that, is a class one graph if is even and is a class two graph if is odd, and also every regular graph of odd order is a class two graph. The next two results is about graphs whose are class one.

Theorem 1.1

 Every bipartite graph is a class one graph.

Corollary 1.2

 If is a graph in which no two vertices of maximum degree are adjacent, then is a class one graph.

A proper vertex labeling of a graph is a function , such that for every pair and of adjacent vertices of . If , then is called a proper -vertex labeling of . The minimum positive integer for which G has a proper -vertex labeling is called the chromatic number of and is denoted by .

A labeling of , , is said to be -distinguishing, if no non-trivial automorphism of preserves all of the vertex labels. The point of the labels on the vertices is to destroy the symmetries of the graph, that is, to make the automorphism group of the labeled graph trivial. Formally, is -distinguishing if for every non-trivial , there exists in such that . Authors often refer to a labeling as a coloring, but there is no assumption that adjacent vertices get different colors. Of course the goal is to minimize the number of colors used. Consequently the distinguishing number of a graph is defined by

 D(G)=min{r| G \textslhasalabelingthatis$r$−distinguishing}.

This number has defined in . If a graph has no nontrivial automorphisms, its distinguishing number is . In other words, for the asymmetric graphs. The other extreme, , occurs if and only if . Collins and Trenk  defined the distinguishing chromatic number of a graph for proper labelings, so is the least number such that has a proper labeling with labels that is only preserved by the trivial automorphism. Similar to this definitions, Kalinowski and Pilśniak  have defined the distinguishing index of which is the least integer such that has an edge coloring with colors that is preserved only by a trivial automorphism. The distinguishing index and number of some examples of graphs was exhibited in [1, 8]. For instance, for every , and for , for . It is easy to see that the value can be large. For example and , for . A symmetric tree, denoted by , is a tree with a central vertex , all leaves at the same distance from and all the vertices which are not leaves with degree . A bisymmetric tree, denoted by , is a tree with a central edge , all leaves at the same distance from and all the vertices which are not leaves with degree . The following theorem gives upper bounds for based on the maximum degree of .

Theorem 1.3

[8, 10]

1. If is a connected graph of order , then , unless is , or .

2. Let be a connected graph that is neither a symmetric nor a bisymmetric tree. If , then , unless is or .

Also, Kalinowski and Pilśniak  defined the distinguishing chromatic index of a graph as the least number such that has a proper edge labeling with labels that is preserved only by the identity automorphism of .

Theorem 1.4

 If is a connected graph of order , then , except for four graphs of small order , , , .

This theorem immediately implies the following interesting result. A proper edge labeling of with colors is called minimal.

Theorem 1.5

 Every connected class 2 graph admits a minimal edge labeling that is not preserved by any nontrivial automorphism.

We need the following results.

Theorem 1.6

 If is a tree of order , then . Moreover, equality is achieved if and only if is either a symmetric or a path of odd length.

Theorem 1.7

 If is a tree of order , then , if and only if is a bisymmetric tree.

In the next section, we compute the distinguishing chromatic index of certain graphs such as friendship and book graphs. More precisely, we present a table of results that shows the chromatic index, the distinguishing index and the distinguishing chromatic index for various families of connected graphs. Also we obtain a relationship between the chromatic distinguishing number of line graph of graph and the chromatic distinguishing index of . In Section 3, we study the distinguishing chromatic index of join and corona product of two graphs.

2 The distinguishing chromatic index of certain graphs

Observation 2.1
1. For any graph , .

2. If has no non-trivial automorphisms, then and . Hence can be much larger than .

By this observation and Theorem 1.4 we can conclude that for any connected graph of order and maximum degree , is or , except for , , , . In the latter case, . Hence, for any connected graph we have , and equality is only achieved for , , , and . By Theorem 1.5, it can be seen that , for class 2 graphs. In the following theorem we present a family of class 1 graphs such that .

Theorem 2.2

Let be a class one graph, i.e., . If there exists a vertex of for which for all automorphisms of , then .

Proof. By contradiction suppose that . Then, for any proper -labeling of , there exists a nonidentity automorphism of preserving the labeling . By hypothesis, we have . Since the incident edges to have different labels, so fixes every adjacent vertex to , because preserves the labeling . By the same argument for every adjacent vertex to , we can conclude that is the identity automorphism, which is a contradiction.

By Theorems 1.3, 1.4 and 1.7, we can characterize all connected graph with .

Theorem 2.3

Let be a connected graph of order and maximum degree .

1. There is no connected graph with .

2. , if and only if .

3. , if and only if is a tree which is not a bisymmetric tree.

Here, we want to obtain the distinguishing chromatic index of complete bipartite graphs. Before we obtain the distinguishing chromatic index of complete bipartite graphs we need the following information of : A labeling with labels of the edges of a complete bipartite graph having parts of size and of size corresponds to a matrix with entries from . The entry of the matrix is whenever the edge between the th vertex in and the th vertex in has label . We call this the bipartite adjacency matrix. For edge labeled complete bipartite graphs, the parts and map to themselves if . In this case, if is the bipartite adjacency matrix, then an automorphism corresponds to selecting permutation matrices and such that . If then we also have automorphisms of the form . For any matrix with entries from the degree of a column is a -tuple with equal to the number of entries that are in the column.

Theorem 2.4

 Let be the adjacency matrix of a -edge labeled complete bipartite graph.

1. If there are two identical rows in , then is not an identity labeling.

2. If is not square and if the columns of have distinct degrees and the rows are distinct, then is an identity labeling. If is square, has distinct rows, distinct column degrees and the multiset of column degrees is different from the multiset of row degrees then is an identity labeling.

Theorem 2.5

The distinguishing chromatic index of complete bipartite graph where , is .

Proof. Let be the following adjacency matrix of a -edge labeled complete bipartite graph,

 A=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝123⋯s−1ss12⋯s−2s−1s−1s1⋯s−3s−2⋮⋮⋮⋮⋮⋮tt+1t+2⋯t−2t−1⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

Then it is clear that is a proper edge labeling. By Theorem 2.4 (ii), it can be concluded that is a distinguishing labeling. In fact, the rows of are distinct, and since the number of label in the th column is one and in the th column, , is zero, so the columns have distinct degrees. Hence .

Before we prove the next result, we need the following preliminaries: By the result obtained by Fisher and Isaak  and independently by Imrich, Jerebic and Klavžar  the distinguishing index of complete bipartite graphs is as follows.

Theorem 2.6

[4, 7] Let be integers such that and . Then

 D′(Kp,q)={rif  q≤rp−⌈logrp⌉−1,r+1if  q≥rp−⌈logrp⌉+1.

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

The friendship graph can be constructed by joining copies of the cycle graph with a common vertex.

Theorem 2.7

 Let . For every ,

 D′(Fn)=⌈13(an)13+13(an)13+13⌉.

The -book graph is defined as the Cartesian product . We call every in the book graph , a page of . The distinguishing index of Cartesian product of star with path for and is , unless and for some integer . In the latter case , (). Since , using this equality we obtain the distinguishing index of book graph .

Theorem 2.8

The entries in Table 1 are correct.

Proof. The chromatic index and the distinguishing index for the classes of graphs given in this table are well-known, we justify the entries in the last column.

Paths of even order. The labeling that uses label 2 for one end-edge and label for the remaining edges is distinguishing, however, it is not a proper labeling and any proper labeling using two labels is not distinguishing, so . A -labeling that is proper and distinguishing is achieved by using for an end-edge and alternating 2’s and 3’s for the remaining edges, thus . Thus the result follows.

Cycles of even order , where . Let the consecutive edges of be . Using label 3 for edges and , label 2 for edges where is odd and label 1 for edges where is even, we get for . All proper 2-labelings of edges of have label preserving automorphisms, thus for all . Therefore , where .

Friendship and book graphs. It is clear that each proper -labeling of edges of is distinguishing and so , by Theorem 2.2. For the book graph , we present a proper -distinguishing edge labeling. Let and be two vertices of degree of , and be the adjacent vertices to , and be the adjacent vertices to , such that for are pages of . We label the edges , and with labels , and mod , respectively, for any , . Also, we label the edge with label . It can be seen that our labeling is a proper -distinguishing labeling.

Complete graphs of even order. It is known that we can partition the edge set of to sets, each set contains an -element perfect matching, say . If we label the edges of -element perfect matching with label , for any , then it can be seen that this labeling is proper. We claim that this labeling is distinguishing. If is an automorphism of preserving the labeling, then fixes the set , for any , setwise. If is a nonidentity automorphism, then without loss of generality we can assume that . Since preserves the labeling so . We can suppose that there exists a vertex of such that the edges and are not in the same perfect matching. Thus the labels of edges and is different, while maps these two edges to each other, which is a contradiction. Then, the identity automorphism is the only automorphism of preserving the labeling, and hence the proper edge labeling is distinguishing, and so .

Complete bipartite graph , . We can partition the edge set of to perfect matching , each of contains edges. For every value of , the labeling that uses label for all edges in , , is a proper labeling. Also, every proper labeling of partitions the edges of to sets such that each of is a perfect matching of and all edges in have the same label, and different from the label of edges in for every where . However, for every proper labeling of we can find a nonidentity automorphism of preserving the labeling, thus . Now the result follows from Theorem 1.4.

In sequel, we want to obtain a relationship between the chromatic distinguishing number of line graph of graph and the chromatic distinguishing index of . For this purpose, we need more information about automorphism group of . For a simple graph , we recall that the line graph is a graph whose vertices are edges of and where two edges are adjacent if they share an endpoint in common. Let be given by for every . In , Sabidussi proved the following theorem which we will use throughout.

Theorem 2.9

 Suppose that is a connected graph that is not , or (see Figure 1). Then is a group isomorphism, and so .

Theorem 2.10

Suppose that is a connected graph of order that is not and . Then .

Proof. First, we show that . For this purpose, let be an edge proper distinguishing labeling of . We define such that where . By the following steps we show that the vertex labeling is proper distinguishing labeling.

1. The vertex labeling is proper. If and are two adjacent vertices of with , then it means that and are incident edges of with , which is impossible. Thus is a proper labeling.

2. The vertex labeling is a distinguishing vertex labeling of , because if is an automorphism of preserving the labeling, then , and hence for any . On the other hand for some automorphism of , by Theorem 2.9. Thus from for any , we can conclude that and so for every . This means that is an automorphism of preserving the labeling , so is the identity automorphism of . Therefore is the identity automorphism of , and hence .

By a similar argument we can prove that , and so the result follows.

Now we state a difference between the distinguishing labeling and chromatic distinguishing labeling of graphs.

Remark 2.11

Despite the distinguishing labeling that for all connected graph , it may be happen that . For instance, let be the graph obtained from by replacing each edge with a path of length three. It can be computed that , while , see figure 2.

We end this section by proposing the following problem.

Problem 2.12

Characterize all connected graphs with .

3 Results for join and corona products

In this section we study the distinguishing chromatic index of join and corona product of graphs. We start with join of graphs. The graph is the join of two graphs and , if and and denoted by .

Theorem 3.1

If and are two connected graphs of orders , respectively, then

 max{Δ(H)+n,Δ(G)+m}≤χ′D(G+H)≤max{χ′D(G),χ′D(H)}+χ′D(Kn,m).

Proof. To prove the left inequality, it is sufficient to know that and . For the right inequality, we first set . We label the edge set of graph (resp. ) with labels (resp. ) in a proper distinguishing way. If and , then we label the middle edges , exactly the same as a proper distinguishing labeling of the complete bipartite graph with labels . Since the graphs and have a proper labeling, so this labeling of edges of is proper, regarding to the label of middle edges. To show this labeling is distinguishing, we suppose that is an automorphism of preserving the labeling. Then, with respect to the label of middle edges, it can be concluded that the restriction of to the vertices of (resp. ) is an automorphism of (resp. ) preserving the labeling. Since the graphs and have been labeled distinguishingly, so the restriction of to vertices of and is the identity automorphism of and , respectively. Therefore, this labeling is distinguishing.

The lower and upper bound of Theorem 3.1 are sharp. For example, we consider where , with . In fact, we can label the edge set of and with labels 1 and 2 in a proper distinguishing way, and label the remaining incident edges to each vertex of with distinct labels such that the incident edges to each vertex of in have distinct labels.

Now we want to obtain a lower and upper bound for the distinguishing chromatic index of corona product. The corona product of two graphs and is defined as the graph obtained by taking one copy of and copies of and joining the -th vertex of to every vertex in the -th copy of . If and be two connected graphs such that , then there is no vertex in the copies of which has the same degree as a vertex in . Because if there exists a vertex in one of the copies of and a vertex in such that , then . So we have , which is a contradiction. Hence, we can state the following lemma.

Lemma 3.2

Let and be two connected graphs such that . If is an arbitrary automorphism of , then the restriction of to the vertices of copy (resp. ) is an automorphism of (resp. ).

Theorem 3.3

If and are two connected graphs of orders , respectively, then

 max{Δ(G)+m,Δ(H)+1}≤χ′D(G∘H)≤max{χ′D(G),χ′D(H)}+m.

Proof. The proof of the left inequality is exactly the same as Theorem 3.1. To prove the right inequality, we first set . We label the edge set of graph (resp. ) with labels (resp. ) in a proper distinguishing way. If and , then we label the middle edge with label for any and . Since the graphs and have been labeled properly, so this labeling of edges of is proper, due to the label of middle edges. To show this labeling is distinguishing, we suppose that is an automorphism of preserving the labeling. Then, by Lemma 3.2, it can be concluded that the restriction of to the vertices of (resp. ) is an automorphism of (resp. ) preserving the labeling. Since the graphs and have been labeled distinguishingly, so the restriction of to vertices of and is the identity automorphism of and respectively. Therefore, this labeling is distinguishing.

The lower and upper bound of Theorem 3.3 are sharp. For instance, we consider where . It can be easily computed that . Now, since and , so the bounds of Theorem 3.3 are sharp.

We end this paper by a remark on the distinguishing chromatic index of join and corona product of graphs and where or .

Remark 3.4
1. If and are two connected graphs such that or , then the maximum degree of is or , respectively, and so , except for . In the latter case, .

2. It is clear that , and so . If , then , and so .

Footnotes

1. Corresponding author

References

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