On Fibonacci numbers and edge coloured trees

An edge-coloring of a graph G is an assignment of colors or labels to the edges of the graph. If {A,B} is a set of two colors used to color the edges of G, then a B-monochromatic subgraph of G is a subgraph H of G induced by B-colored edges of G. We say that G is (A 2B)-edge colored if the set of edges of every maximal B-monochromatic subgraph H of G can be partitioned into pairs of edges which induce a path of length two. If F = fG(1) G(2) : : : G(p)g is the set of all (A 2B)- edge colorings of G, denote by (G(i)) the number of ways of partitioning the edges of G(i) into paths of length two. We de ne the graph parameter (A,2B)(G) as follows: (A 2B)(G) = p∑ i=1 (G(i)): In this study, the researchers will present a detailed exposition of the article, entitled \On Fibonacci Numbers in Edge-Colored Trees by Bednarz, et al. which it discusses the values of the parameter (A 2B)(G) for a special class of trees called tripod and which is denoted by T(m p t). Some relationships between (A 2B)(G) and the Fibonacci numbers Fm will also be discussed and illustrated. Specifically, it will be shown that the second smallest value of (A 2B)(G) for trees is attained when the tree is a tripod.