Every connected graph embeds isometrically into an abelian Cayley host of order at least max(n, 2 diam), binary dimension at least max(diam, ceil(log2 n)), with exact values for stars and odd cycles and a census showing 57% of small graphs gain from non-binary hosts.
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Dimension and Order Bounds for Isometric Embeddings of Graphs into Abelian Cayley Graphs, and the Abelian Dividend
Every connected graph embeds isometrically into an abelian Cayley host of order at least max(n, 2 diam), binary dimension at least max(diam, ceil(log2 n)), with exact values for stars and odd cycles and a census showing 57% of small graphs gain from non-binary hosts.