Abstract

We give a total graph interpretation of the numbers of the Fibonacci type. This graph interpretation relates to an edge colouring by monochromatic paths in graphs. We will show that it works for almost all numbers of the Fibonacci type. Moreover, we give the lower bound and the upper bound for the number of all -edge colourings in trees.

1. Introduction and Preliminary Results

For general concepts about combinatory graph theory and online encyclopedia of integer sequences, see, for example, [1], [2], and [3], respectively. By numbers of the Fibonacci type we mean numbers defined recursively by the th-order linear recurrence relation of the formfor , where and , , are integers and are given integers. For special values of and , , the equality (1) gives the well-known recurrences which define the numbers of the Fibonacci type. They are listed below.(a)Fibonacci numbers :(b)Lucas numbers :(c)Pell numbers :(d)Pell-Lucas numbers :(e)Jacobsthal numbers :(f)Jacobsthal-Lucas numbers :(g)Padovan numbers :(h)Perrin numbers :(i)Tribonacci numbers of the first kind :(j)Tribonacci numbers of the second kind :

These numbers are intensively studied in the literature; they have many interesting interpretations also in graphs. The graph interpretation of the Fibonacci numbers was initiated by Prodinger and Tichy in [4]. In that paper, among others, they showed connections between the Fibonacci and the Lucas numbers and the number of all independent sets in special graphs. Let be the number of all independent sets in a graph . By , , , and we denote a path, a cycle, a tree, and a star of size for , respectively. Then, and ; for details, see [4]. This simple observation gave an impetus for studying the graph parameter in different classes of graphs and their products, from the pure mathematical point of view. This interest was multiplied by the fact that the parameter of a molecular graph was introduced to the combinatorial chemistry by showing some relations between and some physicochemical properties of chemical compounds. For these reasons, the parameter is intensively studied in graphs.

Theorem 1 (see [4]). Let be an integer. For a tree of size , holds. Moreover, iff and iff

The graph parameter relates to other numbers of the Fibonacci type. For Jacobsthal numbers and Jacobsthal-Lucas numbers , the following has been proved.

Theorem 2 (see [5, 6]). Let be an integer. Then, and .

For other results related to the parameter and their applications, see the last survey [7]. In this survey the authors collect and classify the results concerning the graph parameter , most of which are obtained quite recently. Actually for the chemical applications, the index is named as the Merrifield-Simmons index.

Considering the Fibonacci numbers and numbers of the Fibonacci type, we can collect other graph parameters related to the numbers of the Fibonacci type.

Let be the number of all matchings of . Then, and , where is the corona of two graphs. The index is well-known by the Hosoya index. Let be the number of all independent sets in including the set of leaves as a subset. Then, . For this graph parameter, see more details in [8].

Theorem 3 (see [8]). Let be an integer. Then, for a tree of size , holds. Moreover, is the extremal graph achieving the maximum value of .

For the classical Fibonacci numbers and numbers of the Fibonacci type, there are many generalizations with respect to one or more parameters. We list some of these generalized numbers of the Fibonacci type. Let and be integers. We have the following.(1)-generalized Fibonacci numbers (Miles Jr. [9]): with for and .(2)Generalized Fibonacci numbers (Kwaśnik and Włoch [10]): with for .(3)-Fibonacci numbers (Falcón and Plaza [11]): with and .(4)Generalized Pell numbers (Włoch [12]): with and for and , for .(5)-Lucas numbers (Falcon [13]): with and .(6)-Pell numbers (Catarino [14]): with and .(7)-Pell-Lucas numbers (Catarino and Vasco [15]): with .(8)Generalized Lucas numbers (Włoch [16]): with for .(9)Distance Pell numbers (Szynal-Liana and Włoch [5]): with and for , , and for , .(10)Distance companion Pell numbers (Szynal-Liana et al. [6]): with , , and for .(11)Distance Jacobsthal numbers (Szynal-Liana et al. [6]): with and for .(12)Distance Jacobsthal-Lucas numbers (Szynal-Liana et al. [6]): with and for .(13)-distance Fibonacci numbers of the second kind (Bednarz et al. [17]): with if is odd and , if is even and , if , if is odd and , and if is even.(14)-distance Lucas numbers of the second kind (Bednarz et al. [17]): with and for , , , and if is odd and if is even, if is odd and , if is even, if is odd and , and if is even and .

For most of these numbers also some graph interpretations with respect to distance independent sets or matchings were studied (see, e.g., [5, 6, 8, 10, 12, 18–20]).

2. Main Results

The main purpose of this section is to give a total graph interpretation for numbers of the Fibonacci type with respect to a special edge colouring of some graphs.

Let be an undirected, connected, simple graph. Let , , and let , . In particular, can be empty (then we put ). Let be a nonempty family of colours, where for . The set will be called the set of shades of the colour . Consequently, for all , , , holds and this implies that the family has exactly colours.

A graph is ; , -edge coloured by monochromatic paths if for every maximal -monochromatic subgraph of , where , , , there is a partition of into edge disjoint paths of the length . Clearly, if then (; , )-edge colouring by monochromatic paths always exists. With this type of edge colouring of a graph, we associate the following graph parameter.

Let be a graph which can be (; , )-edge coloured by monochromatic paths. Let be a family of distinct (; , )-edge coloured graphs obtained by colouring of a graph and where , , denotes a graph obtained by (; , )-edge colouring by monochromatic paths of a graph .

For (; , )-edge coloured graph by , we denote the number of all partitions of for . Let

Considering the th-order linear recurrence relation (1), we will show that there is a connection between this recurrence equation and the parameter for a special graph .

Theorem 4. Let and be integers. Then,

Proof. Assume that and let us consider (; , )-edge colouring by monochromatic paths of the graph with the numbering of its edges in the natural fashion.
Let us denote by the number of all (; , )-edge colourings by monochromatic paths of the graph , such that the last edge of is coloured by the colour from the fixed set , where . It is clear that If the edge is coloured by one of the shades of colour , that is, by , , , then, according to the definition of (; , )-edge colouring by monochromatic paths, the number is the number of all (; , )-edge colourings by monochromatic paths of the graph . Taking into account that we consider shades of every colour , we getwhich ends the proof.

For the special case of , we can prove the following.

Theorem 5. Let and be integers. Then,(1) for ,(2) for ,(3) for ,(4) for ,(5) for ,(6) for and ,(7) for and ,(8) for and ,(9) for and ,(10) for and ,(11) for and ,(12) for and ,(13) for and .

Proof. Let be as in each statement of the theorem. For the initial terms in each case, we determine the number by inspection. We will analyze some cases.
Consider -edge colouring by monochromatic paths of . If then the unique edge of is coloured by colour . If then we have a path with two edges. So there are exactly two -edge colourings of in the first case using colour and in the second case using the colour . In each case, the graph is monochromatic. Then, and .
Now consider -edge colouring by monochromatic paths of . If then there are exactly two -edge colourings of . The unique edge can be coloured either by or by . Thus, . Let . Then, using only colours and we can colour edges of the graph as follows: , , , . Moreover, there is the unique colouring of the graph using the colour . Then, is -monochromatic. Consequently, .
Consider -edge colouring by monochromatic paths of . If , then there is a unique -edge colouring using only colour . Let , where . Then, in the path , one -monochromatic subpath can exist at most or is -monochromatic. Because -monochromatic path can be chosen on ways, .
In the same way, we can verify the initial conditions in the remaining cases and .
By the initial conditions and by Theorem 4 the result follows.

Analogously as for paths we can prove the following theorem for the cycle .

Theorem 6. Let and be integers. Then,

Proof. Let be a cycle of a size with the numbering of its edges in the natural fashion. Let denote the number of all (; , )-edge colourings by monochromatic paths of such that the edge has the colour from the fixed , where . Clearly, If the edge is coloured by the colour , where and , then The factor follows from the fact that the edge can belong to an -monochromatic path (of length ) on ways, .
Moreover, in the set , we have shades of the colour and can be chosen on ways; thus, which ends the proof.

Using the above theorem, we can give a graph interpretation for the cyclic version of the Fibonacci type. Firstly, we recall some identities given in [6, 13, 16, 17] which will be useful to prove the next theorem.

For and ,(see [13]). For and ,(see [16]). For and ,(see [6]). For , , and ,(see [6]). For and ,(see [17]).

To prove the next theorem, we need the following lemma.

Lemma 7. Let and be integers. Then,

Proof (by induction on ). If , then . Assume that formula (40) holds for an arbitrary . We will prove it for . By the recurrence definitions of the numbers and and by induction hypothesis, we havewhich ends the proof of Lemma 7.

Theorem 8. Let and be integers. Then,(1), for ,(2), for ,(3), for ,(4), for and ,(5), for and ,(6), for and ,(7), for and ,(8), for and ,(9), for and .

Proof. Let and be as in each statement of the theorem.
From Theorem 6, we have that Moreover, by Theorem 5, by the recurrence relation of Fibonacci numbers and the well-known identity for Lucas numbers follows which ends the proof of .
From Theorems 5 and 6, by the recurrence relation of Jacobsthal numbers and the well-known identity for Jacobsthal numbers, we obtainwhich ends the proof of .
Analogously, like in the previous cases, using Theorems 5 and 6 and Lemma 7 and applying known identities for considered numbers, we prove the other conditions such as in what follows.
     +  .
    +   by (35).
     =   by Lemma 7.
by (36), which ends the proof of .
 =  by (37), which ends the proof of .
  +    =   by (38), which ends the proof of .
  =   by (39), which ends the proof of .
Thus, the theorem is proved.

3. -Edge Colouring in Trees

We can study (; , )-edge colouring by monochromatic paths in distinct classes of graphs. For arbitrary and , , the problem seems to be difficult but some interesting results can be obtained for fixed and , . Assume that and . Then, the -edge colouring always exists in an arbitrary graph . In this section, we consider the number of all -edge colourings in trees.

Theorem 9. Let be a tree of size , . Then, Moreover, forand for

Proof. Let be a tree of size , . Firstly, we will show that We prove it by induction on . If then is isomorphic to and the result is obvious. Let be a leaf incident with the edge such that and is isomorphic to either or . Such leaf always exists in a tree, by the basic tree properties. This means that is a tree. Since isolated vertices are not taken into consideration in -edge colourings, it suffices to consider the tree .
Assume that , ,  and are as above and consider the following possibilities.
(i)   has the colour
Then, the subgraph is isomorphic to and has the -edge colouring. By our assumption, there is at least such -edge colourings of in this case.
(ii)   has the colour
Then, by the definition of the -edge colouring, it immediately follows that there is a -monochromatic subgraph with a partition into -monochromatic paths of length .
Let be a maximal -monochromatic subgraph with a partition . Let have the colour and the path belongs to . Then, the subgraph also has -edge colouring and . Clearly, the edge belongs to at least one partition of the -monochromatic subgraph of into -monochromatic paths of length 2. Consequently, if has the colour , then the number of all -edge colourings of is greater than or equal to the number . Since has two isolated vertices, it suffices to consider , so by the induction hypothesis. From the above, . By Theorem 5, the equality is obvious.
Now we show that Since is a connected graph, the maximum value is attained if both edges are adjacent. This means that is the star. Then, there is a maximum number of substars with even number of edges. Consequently, it suffices to calculate the number of all substars of a star with even number of edges. Since and we choose even number of edges, we have at least possibilities of choosing of subset of even edges. Moreover, edges can be partitioned into ways. Additionally, can be -monochromatic.
All this together gives that . The proof of equalities is obvious.

4. Concluding Remarks

Studying the parameter , we can find the graph interpretation with respect to (; , )-edge colourings by monochromatic paths for other numbers of the Fibonacci type not considered in this paper. For some of them, such as the Tribonacci numbers , it does not work in paths and cycles. We can ask about the existence of a graph for which . It is also interesting to consider problems of determining the parameter in distinct classes, clearly for a special (; , )-edge colouring by monochromatic paths.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors wish to thank the referee for all suggestions which improved this paper.