On Frankl and Füredi’s conjecture for 3-uniform hypergraphs
On Frankl and Füredi’s conjecture for 3-uniform hypergraphs
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DOI：
10.1007/s10255-015-0513-1
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发表时间：
2012-11
期刊：
影响因子：
--
通讯作者：
Qingsong Tang;Hao Peng;Cailing Wang;Yuejian Peng
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文献类型：
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作者：
Qingsong Tang;Hao Peng;Cailing Wang;Yuejian Peng
Frankl和f<s:1> redi在[1]中推测，取N(r)的colex排序的前m个集合形成的有m条边的ther-graph具有所有有m条边的r-graph中最大的拉格朗日量。用cr表示这个图，用λ(G)表示超图的拉格朗日。本文首先证明了如果$$\leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3 \end{array}} \right)$$是一个有m条边的左压缩3-图，在顶点集[t]上，最小共轭序为inGcis (t−2−i)(t−2)t的三重图，则λ(G)≤λ(C3,m)。作为一个暗示，Frankl和f<s:1> redi的猜想对于$$ \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right) - 6 \leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right)$$是正确的。
Frankl and Füredi in [1] conjectured that ther-graph with m edges formed by taking the first m sets in the colex ordering of N(r)has the largest Lagrangian of allr-graphs with m edges. Denote thisr-graph byCr,mand the Lagrangian of a hypergraph byλ(G). In this paper, we first show that if $$\leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3 \end{array}} \right)$$,Gis a left-compressed 3-graph with m edges and on vertex set [t], the triple with minimum colex ordering inGcis (t− 2 −i)(t− 2)t, thenλ(G) ≤λ(C3,m). As an implication, the conjecture of Frankl and Füredi is true for $$ \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right) - 6 \leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right)$$.