4 problems

Let $G$ be a graph with maximum degree $\Delta(G)\geq 2$. For positive integers $m,n$, $G^{\frac{m}{n}}$ denotes the $m$-power of the $n$-subdivision of $G$; in particular,…

Let $G$ be a graph with maximum degree $\Delta(G)\geq 3$, and let $m,n$ be positive integers with $\frac{m}{n}\in\mathbb{Q}\cap(0,1)$. The graph $G^{\frac{m}{n}}$ is the $m$-power…

For positive integers $m,n$ with $1<m<n$, let $G^{\frac{m}{n}}$ be the $m$th power of the $n$-subdivision of a graph $G$, and let $\chi$ and $\omega$ denote its chromatic and cliqu…

Let $G$ be a connected graph, let $\Delta(G)$ denote its maximum degree, and for positive integers $m,n$ let $G^{\frac{m}{n}}$ be the $m$th power of the $n$-subdivision of $G$. Ira…