AI Networking Cookbook: Practical recipes for AI-assisted network automation and development
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We prove that every $n$-vertex $15$-tough graph contains a cycle of length $\Omega(n^{1/20})$.
Let $r \ge 2$ and $t \ge 1$. It is shown that if $T$ is an $r$-uniform tight tree with $t$ edges and $H$ is a $T$-free $r$-uniform hypergraph, then $|E(H)|\le (t-1)|\sh H|/r$, where $\sh H$ is the $(r-1)$-shadow of $H$. \iffalse Equality holds only for $(n,t + r - 2,r)$-designs.\fi The bound is tight infinitely often,...
For every real $t>0$, we prove that every $2$-connected $t$-tough graph contains a cycle of length at least $\ell$ whenever $n \leq \ell(1 + 1/t)^{\lfloor \ell/2\rfloor - 1}$. This establishes the logarithmic bound conjectured by Broersma, van den Heuvel, Jung, and Veldman.
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