Fibonacci-run graphs I: basic properties

TL;DR

This paper explores Fibonacci-run graphs, a new class of interconnection networks derived from Fibonacci-related binary strings, analyzing their structural properties and connectivity features.

Contribution

It introduces Fibonacci-run graphs, investigates their basic properties, and compares them to Fibonacci cubes, highlighting differences in edges and connectivity.

Findings

Fibonacci-run graphs have fewer edges than Fibonacci cubes.

They exhibit Hamiltonicity and specific degree sequences.

The paper derives formulas for edges and explores connectivity properties.

Abstract

Among the classical models for interconnection networks are hypercubes and Fibonacci cubes. Fibonacci cubes are induced subgraphs of hypercubes obtained by restricting the vertex set to those binary strings which do not contain consecutive 1s, counted by Fibonacci numbers. Another set of binary strings which are counted by Fibonacci numbers are those with a restriction on the runlengths. Induced subgraphs of the hypercube on the latter strings as vertices define Fibonacci-run graphs. They have the same number of vertices as Fibonacci cubes, but fewer edges and different connectivity properties. We obtain properties of Fibonacci-run graphs including the number of edges, the analogue of the fundamental recursion, the average degree of a vertex, Hamiltonicity, special degree sequences, and the number of hypercubes they contain. A detailed study of the degree sequences of Fibonacci-run graphs is interesting in its own right and is reported in a companion paper.