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John Talbot (2002)
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Hence λ(G) = λ(G, x) ≤ λ(G ′ , z) = λ(G ′ )
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If Case 6 and a ≥ 7 happen, then by Lemma 3.6, λ(G) ≤ λ(C 3,m )
) − E. Since G is leftcompressed
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) − E = {(t − 3)(t − 2)t, (t − 4)(t − 2)t, (t − 3)(t − 2)(t − 1)}. If Cases 1, 2, 4, 5 happen, then by Lemma 3.3, λ(G) ≤ λ(C 3,m )
Qingsong Tang, Yuejian Peng, Xiangde Zhang, Cheng Zhao (2014)
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Frankl and Füredi in [1] conjectured that the r-graph with m edges formed by taking the first m sets in the colex ordering of N(r) has the largest Lagrangian of all r-graphs with m edges. Denote this r-graph by C r,m and the Lagrangian of a hypergraph by λ(G). In this paper, we first show that if $$\leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3 \end{array}} \right)$$ , G is a left-compressed 3-graph with m edges and on vertex set [t], the triple with minimum colex ordering in G c is (t − 2 − i)(t − 2)t, then λ(G) ≤ λ(C 3,m ). As an implication, the conjecture of Frankl and Füredi is true for $$ \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right) - 6 \leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right)$$ .
Acta Mathematicae Applicatae Sinica – Springer Journals
Published: Apr 5, 2016
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