Goddyn, Luis A.

Circular colouring the orthogonality graph ★★

Author(s): DeVos; Ghebleh; Goddyn; Mohar; Naserasr

Let ${\mathcal O}$ denote the graph with vertex set consisting of all lines through the origin in ${\mathbb R}^3$ and two vertices adjacent in ${\mathcal O}$ if they are perpendicular.

\begin{problem} Is $\chi_c({\mathcal O}) = 4$? \end{problem}

Keywords: circular coloring; geometric graph; orthogonality

A conjecture on iterated circumcentres ★★

Author(s): Goddyn

\begin{conjecture} Let $p_1,p_2,p_3,\ldots$ be a sequence of points in ${\mathbb R}^d$ with the property that for every $i \ge d+2$, the points $p_{i-1}, p_{i-2}, \ldots p_{i-d-1}$ are distinct, lie on a unique sphere, and further, $p_i$ is the center of this sphere. If this sequence is periodic, must its period be $2d+4$? \end{conjecture}

Keywords: periodic; plane geometry; sequence

(2 + epsilon)-flow conjecture ★★★

Author(s): Goddyn; Seymour

\begin{conjecture} For every $\epsilon>0$ there exists an integer $k$ so that every $k$-\Def[edge-connected]{connectivity (graph theory)} graph has a $(2+\epsilon)$-flow. \end{conjecture}

Keywords: edge-connectivity; flow

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