Fibonacci-run graphs II: Degree sequences

Abstract

Fibonacci cubes are induced subgraphs of hypercube graphs obtained by restricting the vertex set to those binary strings which do not contain consecutive 1s. This class of graphs has been studied extensively and generalized in many different directions. Induced subgraphs of the hypercube on binary strings with restricted runlengths as vertices define Fibonacci-run graphs. These graphs have the same number of vertices as Fibonacci cubes, but fewer edges and different graph theoretical properties. Basic properties of Fibonacci-run graphs are presented in a companion paper, while in this paper we consider the nature of the degree sequences of Fibonacci-run graphs. The generating function we obtain is a refinement of the generating function of the degree sequences, and has a number of corollaries, obtained as specializations. We also obtain several properties of Fibonacci-run graphs viewed as a partially ordered set, and discuss its embedding properties.