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The exact isoperimetric inequality for ternary and quaternary cubes
Slivnik, Tomaž, 1969-
We extend the well-known edge-isoperimetric inequality of Harper, Bernstein and Hart to ternary and quaternary cubes. More generally, let ▫$Q$▫ be the graph with vertex set ▫$V = \prod_{i=1}^n ...\lbrack k_i \rback$▫ in which ▫$x \in V$▫ is joined to ▫$y \in V$▫ if for some ▫$i$▫ we have ▫$\|x_i -y_i\| = 1$▫ and ▫$x_j=y_j$▫ for all ▫$j \ne i$▫. If ▫$k_1 \ge ...\ge k_n$▫ and ▫$k_2 \le 4$▫, we prove that for any ▫$0 \le m \le \|V\|$▫, no ▫$m$▫-set of vertices of ▫$Q$▫ is joined to the rest of ▫$Q$▫ by fewer edges than the set of the first ▫$m$▫ vertices of ▫$Q$▫ in the lexicographic ordering on ▫$V$▫.
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