5 problems

Let $\mathcal{R}_n^l$ be the Lucas-run graph and let $C_{\mathcal{R}_n^l}(x)$ denote its cube polynomial, with $C_{\mathcal{R}_n^l}(x)=\sum_{k\geq 0}c_kx^k$, where $c_k$ is the num…

Let $Q_n$ denote the $n$-dimensional hypercube graph, and let $\mathcal{R}_m$ denote the Fibonacci-run graph of parameter $m$. Hypercube embedding conjecture. The smallest integer…

Let $\mathcal{R}_n$ be the Fibonacci-run graph on the relevant Fibonacci-run strings, and let $k$ be a nonnegative integer. Write the generating function for the number of vertices…

Let $\mathcal{R}_n$ be a Fibonacci-run graph, and let $L_n$ denote the lower bound for its independence number given in Lemma. Independence-bound conjecture. The lower bound $L_n$…

Let $\mathcal{R}_n$ be the Fibonacci-run graph of dimension $n$, and let $\operatorname{diam}(\mathcal{R}_n)$ denote its diameter. Diameter conjecture. … The formula is supported b…