Long-cycle conjecture for graphs of given average degree
Prove that for every cepsilon>0 there exist d_0,cgamma>0 such that every graph G of average degree dcgeq d_0 satisfies that, when pcgeq(1+cepsilon)/d, the percolated graph G_p contains a cycle of length at least cgamma d asymptotically almost surely as |G|ctocinfinity.
References
We reiterate a conjecture of Krivelevich and Samotij (see also ), stating that such a result should still be true if we only assume that the average degree is $d$. \begin{conjecture}[]\label{conj:long cycle} For every $\varepsilon>0$ there exist $d_0,\gamma>0$ such that the following holds. If $G$ is a graph of average degree $d\geq d_0$ and $p\geq (1+\varepsilon)/d$, then $G_p$ contains a cycle of length at least $\gamma d$ a.a.s.\ as $|G| \to \infty$. \end{conjecture}