5 problems

Let $G$ be a graph, and consider the Cartesian square $G\times G$. An ordering of the vertices is optimal for the edge-isoperimetric problem (EIP) when it minimizes the number of b…

Subadditivity-plus-sigma conjecture. For all $n,m,\ell_a,\ell_b\in\mathbb N$ with $m^n\geq\ell_a\geq\ell_b>0$, if $\ell_a+\ell_b\leq m^n$, then

Decorated Sierpinski graph conjecture. For every $\ell$ with $0\leq\ell\leq m^n$,

Let $I=\left\{0,1,\ldots,s-1\right\}$, $J=\left\{s,s+1,\ldots,s+t-1\right\}$, and $K=\left\{s+t,s+t+1,\ldots,m-1\right\}$, where $s,t\geq0$ and $s+t\leq m$. Let $S_{s,t}(n,m)$ be t…

Let $S(n,m)$ be the generalized and expanded Sierpinski graph, with vertex set $V_{S(n,m)}=\left\{0,1,\ldots,m-1\right\}^{n}$. Let $Lex$ denote the lexicographic order on this vert…