Erdős 1007, dimension five: fifteen edges, attained by K6 and K1,3,3
The least number of edges of a graph of Euclidean dimension five (Erdős–Harary–Tutte) is 15, and both K6 and K1,3,3 attain it (Chaffee and Noble 2016). These are formal-conjectures' erdos_1007.variants.dimension_five and dimension_five_extremal. A graph has dimension n when n is the least dimension in which its vertices can be placed injectively with every edge a unit segment. The lower bound is Chaffee and Noble's Theorem 10: every graph with at most fourteen edges has such a placement in ℝ⁴. Its proof removes a vertex of degree at most three, joins that vertex's neighbours to one another,…
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