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arXiv · 1604.05084

On a problem by Shapozenko on Johnson graphs

Víctor Diego·Oriol Serra·Lluís Vena
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Abstract

The Johnson graph $J(n,m)$ has the $m$--subsets of $\{1,2,\ldots,n\}$ as vertices and two subsets are adjacent in the graph if they share $m-1$ elements. Shapozenko asked about the isoperimetric function $\mu_{n,m}(k)$ of Johnson graphs, that is, the cardinality of the smallest boundary of sets with $k$ vertices in $J(n,m)$ for each $1\le k\le {n\choose m}$. We give an upper bound for $\mu_{n,m}(k)$ and show that, for each given $k$ such that the solution to the Shadow Minimization Problem in the Boolean lattice is unique, and each sufficiently large $n$, the given upper bound is tight. We also show that the bound is tight for the small values of $k\le m+1$ and for all values of $k$ when $m=2$.

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Víctor Diego, Oriol Serra, Lluís Vena. 2016-04-18. On a problem by Shapozenko on Johnson graphs. https://arxiv.org/abs/1604.05084

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