On the chromatic polynomial and the domination number of k-Fibonacci cubes

Abstract

Fibonacci cubes are defined as subgraphs of hypercubes, where the vertices are those without two consecutive 1's in their binary string representation. k-Fibonacci cubes are in turn special subgraphs of Fibonacci cubes obtained by eliminating certain edges. This elimination is carried out at the step analogous to where the fundamental recursion is used to construct Fibonacci cubes themselves from the two previous cubes by link edges. In this work, we calculate the vertex chromatic polynomial of k-Fibonacci cubes for k = 1, 2. We also determine the domination number and the total domination number of k-Fibonacci cubes for n, k ≤ 12 by using an integer programming formulation.

Domination type parameters of Pell graphs

“…By using the integer linear programming approach several additional parameters of small Fibonacci cubes, Lucas cubes and k-Fibonacci cubes are obtained in [2,6,12,20]. In this section we use a similar approach to obtain domination type parameters of small Pell graphs.…”

“…Furthermore, upper bounds and lower bounds on domination and total domination numbers of Fibonacci and Lucas cubes are obtained in [2,3,17,18,19,20]. The domination and total domination number of k-Fibonacci cubes are considered in [6]. In this work, we studied some domination type parameters of Pell graphs.…”

On the Cube Polynomials of Padovan and Lucas–Padovan Cubes

“…As it is well known, the Fibonacci cube has become a popular interconnection topology. The Fibonacci cube was first introduced by Hsu [2], and many scholars studied cube polynomial in [1,[3][4][5][6][7][8][9].…”

“…The Lucas-Padovan sequence is defined by the following rules; let LP 1 = 1 and, for n ≥ 2, LP n = P n−1 + P n+1 , where P n is the nth Padovan number. The first few numbers of the Lucas-Padovan sequence LP n , for n ≥ 1, are 1, 2, 3, 3,5,6,8,11,14,19,25,33,44, 58, 77, . .…”