A vertex of degree one is called an end-vertex and the set of end-vertices of G is denoted by End(G). For a positive integer k, a tree T be called k-ended tree if |End(T )| ≤ k. In this paper, we obtain sufficient conditions for spanning k-trees of 3-regular connected graphs. We give a construction sequence of graphs satisfying the condition. At the end, we present a conjecture about spanning k-ended trees of 3-regular connected graphs.
A lattice path L in Z d of length k with steps in a given set S ⊆ Z d , or S-path for short, is a sequence ν 1 , ν 2 , . . . , ν k ∈ Z d such that the steps ν i − ν i−1 lie in S for all i = 2, . . . , k. Let T m,n be the m × n table in the first area of xy-axis and put S = {(1, 0), (1, 1), (1, −1)}. Accordingly, let I m (n) denote the number of S-paths starting from the first column and ending at the last column of T . We will study the numbers I m (n) and give explicit formulas for special values of m and n. As a result, we prove a conjecture of Alexander R. Povolotsky. We conclude the paper with some applications to Fibonacci and Pell-Lucas numbers and posing an open problem.
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