On a conjecture of Eǧecioǧlu and Iršič

Abstract

In 2021, Ö. Eǧecioǧlu, V. Iršič introduced the concept of Fibonacci-run graph $R_n$ as an induced subgraph of Hypercube. They conjectured that the diameter of $R_n$ is given by $n-\lfloor {(1+\frac{n}{2})}^{\frac{1}{2}}-\frac{3}{4}\rfloor$. In this paper, we introduce the concept of distance-barriers between vertices in $R_n$ and provide a lower bound for the diameter of $R_n$ via this concept. By constructing different types of distance-barriers, we show that the conjecture fails to hold for all $n\ge 230$ and for some $n$ between 91 and 229. The lower bounds obtained turn out to be better than the result given in the conjecture.