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Let $G$ denote a finite, connected, simple graph, and let $D$ denote its diameter, and let $d_{G}(u, v)$ denote the distance between vertices $u$ and $v$ in $V(G)$. An antipodal labeling of $G$ is a mapping $f\colon V(G) \to \mathbb{N}_{0}$ such that, for each pair $(u, v)$ consisting of distinct vertices in $V(G)$, th...
Let $S$ denote a $k$-subset of $\mathbb{Z}$. Let $G$ be a (multiplicative) group of size $k$. Assume that there exists a function $\varphi\colon S \to G$ that is injective and that satisfies the following: If $x, y, z \in S$ and $z = xy$, then $\varphi(z) = \varphi(x) \varphi(y)$. In 1990, Forcade and Pollington conjec...
For $a$ in $\mathbb{R}$, a subset $V$ contained in $\mathbb{R}^{n}$ is said to be $a$-convex if $x, y \in V \Longrightarrow a x + (1-a) y \in V$. According to Pinch [Math. Proc. Cambridge Philos. Soc., 1985], the $a$-convex hull of $V$ is the intersection of all of the $a$-convex subsets of $\mathbb{R}^{n}$ that contai...
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