Extremal Trees with respect to Number of (A, B, 2C)-Edge Colourings

2P n−1 +P n−2 for n ≥ 2withP 0 = 0,P 1 = 1.Table 1 includes first terms of the sequence {P n }. 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 havemany different applications and interpretations. Some of them are closely related to the Hosoya index Z(G) (defined as a number of all matchings in the graph G, including the empty matching) and the Merrifield-Simmons index σ(G) (defined as a number of all independent sets in G, including the empty set); see [2] and its references. It is well-known that


Introduction and Preliminary Results
For a general concept, see [1].The Fibonacci sequence {  } is defined recursively by the second-order recurrence relation   =  −1 + −2 for  ≥ 2 with the initial conditions  0 =  1 = 1.A related sequence is the Pell sequence {  } defined by   = 2 −1 + −2 for  ≥ 2 with  0 = 0,  1 = 1.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 (P  ) =   and (P  ∘ K 1 ) =  +1 , for  ≥ 1, where P  is an -vertex path, K  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][4][5][6][7][8][9][10][11][12][13][14][15].
Let {, , } be the set of colours.By (, , 2)-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 (, , 2)-edge colourings of the graph .The following result was given in [10].
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 K  and K 1 ; that is, (K  ∘ K 1 ) =   for  ≥ 1.For more interpretations see [16,17].Another interpretation of the sequence {  } in graphs, which is closely related to (, , 2)-edge colouring of -edge star K 1, , was given in [11].
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 (, , 2)-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.
Let  ∈ () be the vertex of degree  ≥ 1 and let  1 ,  2 , . . .,   ∈ () be neighbours of .By (3) in Theorem 3 we have and by inequality (2) in Theorem 3 we have It should be noted that ∑  =1 (K 1, \ {V 0 V  }) = (K 1,−1 ) and so (8) gives Since  ≥ 2, then from ( 7) and ( 9) we have (  * K V 1, ) < (  * K V 0 1, ), which completes the proof.Theorem 5. Let  ≥ 2 be an integer and let   be a tree with  vertices.Then Proof (by induction on the number of vertices of degree  ≥ 2 in the tree   ).Let  , be an -vertex tree with exactly  vertices of degree  ≥ 2. If  = 1 then the result is obvious, because  ,1 is an -vertex star K 1,−1 .Assume that inequality (10) holds for  , with arbitrary  ≥ 1.We will prove that it holds for  ,+1 .Note that for each tree  ,+1 there exists  ∈ ( ,+1 ), such that  ,+1 is isomorphic to   −+1, * K V 1,−1 , where V is the leaf of the star K 1,−1 and  ≥ 3. Applying Theorem 4 we have where V 0 is the center of the star K 1,−1 .Note that   −+1, * K V 0 1,−1 is the -vertex tree with  vertices of the degree  ≥ 2. Thus, by the induction hypothesis we have ( ,+1 ) ≤ (K 1,−1 ), which completes the proof.Remark 6.From Theorems 4 and 5 we can see that the star K 1,−1 is a unique graph which maximizes the number of (, , 2)-edge colourings in trees of given order .

Colourings in Trees
Now we show that, among all trees with the given number of vertices , the path P  minimizes the number of all (, , 2)-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.

Proof (by induction on 𝑞).
For  = 2 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  ≥ 2. We show that it holds for  + 1; namely, where  +1 is a positive integer and  =  1 + 2 +⋅ ⋅ ⋅+  +1.Using ( 14) and the induction hypothesis we obtain which ends the proof.

Table 1 :
The first terms of {  } and {  }.