6 problems

Let $n$-gons be regular polygons, and write the density of the densest packing with plane group symmetry $G$ as the corresponding densest $G$-packing density. An $n$-gon has a $k$-…

Let $B_1(n)$ denote the unit ball in $\mathbb{R}^n$, let $\mathbb{S}^{n-1}$ denote the unit sphere, and let $L oindent oise$ be a line. For each $x otin K$, let $Cx$ be the cone ci…

Let $G_\alpha(m;1,1,c)$ be the alpha family, and suppose $\gcd(c,m)>1$. Nontrivial-gcd colorability conjecture. The graph $G_\alpha(m;1,1,c)$ is 3-edge-colorable. The paper notes t…

Let $G_\beta(m;a,b,c)$ be the beta family of cyclic graphs constructed in the paper. Beta-family non-snark conjecture. The graph $G_\beta(m;a,b,c)$ is never a snark. The displayed…

Let $T^1_\alpha(m;a,a,c)$, $T^1_\beta(m;a,a,c)$, and $T^2_\alpha(m;a,a,c)$ be the snark families defined from the two triangle-destroying deletions of the Petersen-derived 5-pole.…

Let $S$ be a Loupekine 5-pole, let $x\in\{\alpha,\beta\}$, and let $S_x(m;a,a,c)$ denote the associated cyclic pseudo-Loupekine graph. Assume that $m/\gcd(m,a)$ is even and…