Abstract
We determine the smallest and the largest number of -edge colourings in trees. We prove that the star is a unique tree that maximizes the number of all of the -edge colourings and that the path is a unique tree that minimizes it.
1. Introduction and Preliminary Results
For a general concept, see [1]. The Fibonacci sequence is defined recursively by the second-order recurrence relation for with the initial conditions . A related sequence is the Pell sequence defined by for with , Table 1 includes first terms of the sequence The terms of Fibonacci and Pell sequences are called Fibonacci numbers and Pell numbers, respectively. The numbers of the Fibonacci type play an important role in distinct areas of mathematics and they have many different applications and interpretations. Some of them are closely related to the Hosoya index (defined as a number of all matchings in the graph , including the empty matching) and the Merrifield-Simmons index (defined as a number of all independent sets in , including the empty set); see [2] and its references. It is well-known that and , for , where is an -vertex path, is an -vertex complete graph, and denotes the corona of two graphs. The numbers of the Fibonacci type in the graph theory were studied intensively also in [3–15].
Consider a simple, undirected graph with the vertex set and the edge set In [11] we introduced an -edge colouring of a graph defined in the following way. Let be -edge coloured graph with the set of colours , where Moreover, let be positive integers. We say that a subgraph of is -monochromatic if all its edges are coloured alike by colour The graph is said to be -edge coloured, if every maximal (with respect to set inclusion) -monochromatic subgraph of can be partitioned into edge-disjoint paths of the length , This type of edge colouring of graph generalizes the edge colouring introduced by Piejko and Włoch in [10] and the edge colouring by monochromatic paths introduced by Trojnar-Spelina and Włoch in [13]. Many interesting results concerning some special kinds of -edge colouring of graphs can be found in [10, 11]. We recall some of them.
Let be the set of colours. By -edge colouring we denote the 3-edge colouring of graph , such that every -monochromatic subgraph of can be partitioned into edge-disjoint paths of the even length. Let be the number of all -edge colourings of the graph The following result was given in [10].
Theorem 1 (see [10]). Let be an integer. Then
Let be the sequence defined by the relation for with the initial conditions , We can find a few first terms of in Table 1.
The sequence has many distinct interpretations also in graphs. It is worth mentioning that is the Hosoya index of the corona of the complete graphs and ; that is, for For more interpretations see [16, 17].
Another interpretation of the sequence in graphs, which is closely related to -edge colouring of -edge star , was given in [11].
Theorem 2 (see [11]). Let be a positive integer. Then
In [12] Prodinger and Tichy proved that the star is a tree that maximizes the Merrifield-Simmons index, while the path is a tree that minimizes it. In this paper we obtain an analogous result for the number of -edge colourings in trees.
Let and be given graphs with distinguished vertices and By we denote the graph obtained from and by identifying vertices and (see Figure 1) and by we denote the graph obtained from and by adding the edge (see Figure 2).
For the notation means the graph obtained from by deleting the edge We prove the following.
Theorem 3. Let , and let be neighbours of and let be neighbours of , where and are positive integers. Then Furthermore, if then
Proof. By , , and we denote the number of all -edge colourings of the graph such that an edge has a colour , , or , respectively. It can be easily seen that and are equal to the number of all -edge colourings of the graph , multiplied by the number of all -edge colourings of the graph Moreover, is equal to the number of all -edge colourings of the graph multiplied by the number of all -edge colourings of graphs , where , plus the number of all -edge colourings of the graph multiplied by the number of all -edge colourings of graphs , where In other words, and Since , then we obtain equality (1).
By we denote the edge , where and by we denote the edge , where Assume that and for let be the number of all -edge colourings of the graph with exactly -monochromatic paths , such that and Observe that and It should be noted that and so , which gives inequality (2). Moreover, if then also and we obtain equality (3). This completes the proof.
2. The Largest Number of -Edge Colourings in Trees
In this section we will show that, among all trees with the given number of vertices , the star maximizes the number of -edge colourings. Moreover the star is the unique tree with such property. To prove it we need the following.
Theorem 4. Let be an integer. Then for a graph and arbitrary one has where is the leaf of and is the center of
Proof. Let and Let be the vertex of degree and let be neighbours of By (3) in Theorem 3 we have and by inequality (2) in Theorem 3 we have It should be noted that and so (8) gives Since , then from (7) and (9) we have , which completes the proof.
Theorem 5. Let be an integer and let be a tree with vertices. Then
Proof (by induction on the number of vertices of degree in the tree ). Let be an -vertex tree with exactly vertices of degree If then the result is obvious, because is an -vertex star . Assume that inequality (10) holds for with arbitrary . We will prove that it holds for . Note that for each tree there exists , such that is isomorphic to , where is the leaf of the star and Applying Theorem 4 we have where is the center of the star Note that is the -vertex tree with vertices of the degree . Thus, by the induction hypothesis we have , which completes the proof.
Remark 6. From Theorems 4 and 5 we can see that the star is a unique graph which maximizes the number of -edge colourings in trees of given order .
3. The Smallest Number of -Edge Colourings in Trees
Now we show that, among all trees with the given number of vertices , the path minimizes the number of all -edge colourings and that it is the unique tree with such property. To prove it we need some initial results. First we prove the following property of the Pell numbers.
Theorem 7. Let and be integers for Then where
Proof (by induction on ). For we have We can check the above inequality using the well-known identity for the Pell numbers Assume that inequality (12) holds for an arbitrary We show that it holds for ; namely, where is a positive integer and Using (14) and the induction hypothesis we obtain which ends the proof.
Let , , be positive integers. By we denote the subdivision of a star in such way that in the resulting graph an th edge of is replaced by a path of length , For the graph is a star and for the graph is a path For the star subdivision is called a tripod. In [11] we proved that for all positive integers , , and We will use the following notations:
Theorem 8. Let , , be positive integers. Then
Proof (by induction on ). If then the result we have immediately from (17). Assume that the inequality holds for , with arbitrary . We will prove that it holds for . Note that is isomorphic to , where is the center of , is the leaf of , and is a positive integer. Thus, applying (3) of Theorem 3 we obtain Note that for all By induction hypothesis we have for all and . Therefore (21) and Theorems 1 and 7 give Thus by (20), (22), and Theorem 1 we have By the induction hypothesis we have and so (23) gives Using identity (14) we have , which completes the proof.
Theorem 9. Let , , be positive integers and let . Then for a graph and arbitrary one has where is the center of and is the leaf of the path
Proof. Let be the vertex of degree and let be neighbours of Note that is isomorphic to , where is the graph obtained from by adding a vertex and an edge Therefore, equality (3) of Theorem 3 gives Moreover equality (1) of Theorem 3 gives By (27), Theorem 1, and the definition of we obtain From Theorem 8 we have and and so (26) and (28) give , which ends the proof.
Theorem 10. Let be an integer and let be a tree with vertices. Then
Proof (by induction on the number of vertices of degree in the tree ). Let be the -vertex tree with exactly vertices of degree If then the result is obvious, because is an -vertex path . Assume that inequality (29) holds for with arbitrary We will prove that it holds for Note that for each tree there exists , such that is isomorphic to , where is the center of , , and Applying Theorem 9 we have where is the leaf of the path Note that is the -vertex tree with vertices of degree . Thus by the induction hypothesis we have and the proof is complete.
Remark 11. From Theorems 9 and 10 we can see that the path is a unique graph which minimizes the number of -edge colourings in trees of given order .
From previous theorems we have also the following.
Corollary 12. If is a tree with the number of vertices , then Moreover, if is different from and , then
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.