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The domination number of the graph defined by two levels of the n-cube, II
European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2020-08-15 , DOI: 10.1016/j.ejc.2020.103201
József Balogh , Gyula O.H. Katona , William Linz , Zsolt Tuza

Consider all k-element subsets and -element subsets (k>) of an n-element set as vertices of a bipartite graph. Two vertices are adjacent if the corresponding -element set is a subset of the corresponding k-element set. Let Gk, denote this graph. The domination number of Gk,1 was exactly determined by Badakhshian, Katona and Tuza. A conjecture was also stated there on the asymptotic value (n tending to infinity) of the domination number of Gk,2. Here we prove the conjecture, determining the asymptotic value of the domination number γ(Gk,2)=k+32(k1)(k+1)n2+o(n2).



中文翻译:

图的支配数由两个层次的 ñ-立方体,二

考虑所有 ķ-元素子集和 元素子集 ķ>ñ元素设置为二部图的顶点。如果对应的两个顶点相邻-element set是相应元素的子集 ķ-元素集。让Gķ表示该图。支配数Gķ1个由Badakhshian,Katona和Tuza完全确定。那里也有一个关于渐近值的猜想(ñ 趋于无穷大) Gķ2。在这里我们证明了猜想,确定了控制数的渐近值γGķ2=ķ+32ķ-1个ķ+1个ñ2+Øñ2

更新日期:2020-08-15
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