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07 February 2025
 
  » arxiv » 1605.0450

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Bandwidth of graphs resulting from the edge clique covering problem
Konrad Engel ; Sebastian Hanisch ;
Date 2 May 2016
AbstractLet $n,k,b$ be integers with $1 le k-1 le b le n$ and let $G_{n,k,b}$ be the graph whose vertices are the $k$-element subsets $X$ of ${0,dots,n}$ with $max(X)-min(X) le b$ and where two such vertices $X,Y$ are joined by an edge if $max(X cup Y) - min(X cup Y) le b$. These graphs are generated by applying a transformation to maximal $k$-uniform hypergraphs of bandwidth $b$ that is used to reduce the (weak) edge clique covering problem to a vertex clique covering problem. The bandwidth of $G_{n,k,b}$ is thus the largest possible bandwidth of any transformed $k$-uniform hypergraph of bandwidth $b$. For $bgeq frac{n+k-1}{2}$, the exact bandwidth of these graphs is determined. For $b<frac{n+k-1}{2}$, the bandwidth is asymptotically determined in the case of $b=o(n)$ and in the case of $b$ growing linearly in $n$ with a factor $eta in (0,0.5]$, where for one case only bounds could be found. It is conjectured that the upper bound of this open case is the right asymptotic value.
Source arXiv, 1605.0450
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