Total Graph Interpretation of the Numbers of the Fibonacci Type

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 ( 1 , 2 1 )-edge colourings in trees.

On Fibonacci numbers in edge coloured trees

“…A graph G is (A, 2B)-edge coloured if for every maximal (with respect to set inclusion) B-monochromatic subgraph H of G there exists a partition of H into edge disjoint paths of length 2. We have no restriction on the colour A, so (A, 2B)-edge colouring exists for an arbitrary graph G. It is worth mentioning that the concept of (A, 2B)-edge colouring is a special case of edge shade colouring of a graph (more details for edge shade colouring can be found in [1] ), where 1 ≤ p ≤ l, be the number of all partitions into edge disjoint paths of length 2 of all B-monochromatic subgraphs of G (p) .…”

“…The parameter σ (A,2B) (G) was studied for different classes of graphs (see [1]). We recall the main result for trees.…”

On a four-parameter generalization of some special sequences

“…Finally, in 1982, Prodinger and Tichy [25] gave a complete graph interpretation to the Fibonacci sequence by exhibiting connections between the Fibonacci and Lucas numbers and the number of independent sets in some especial graphs. Some recent progress on interpreting generalizations of the Fibonacci sequence in terms of graphs can be found in [4,24,31].…”

On F(p, n) - Fibonacci quaternions

“…Besides the usual Fibonacci and Lucas numbers many kinds of generalizations of these numbers have been presented in the literature, see their list in [1]. In [11] Kwaśnik and I. W loch introduced the generalized Fibonacci numbers F (p, n) and the generalized Lucas numbers L(p, n) defined as follows F (p, n) = n + 1, for n = 0, 1, .…”

“…Apart Fibonacci and Lucas numbers there are known numbers defined recursively by the second order linear recurrence relations. It is necessary to mention Pell numbers, Pell-Lucas numbers, Jacobsthal numbers, Jacobsthal-Lucas numbers, for details see [1]. These numbers are also named as numbers of the Fibonacci type and they have applications in distinct areas of mathematics.…”