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- W3082020172 abstract "We study the set ${cal L}(G)$ of lengths of all cycles that appear in a random $d$-regular $G$ on $n$ vertices for a fixed $dgeq 3$, as well as in Erdős--Renyi random graphs on $n$ vertices with a fixed average degree $c>1$. Fundamental results on the distribution of cycle counts in these models were established in the 1980's and early 1990's, with a focus on the extreme lengths: cycles of fixed length, and cycles of length linear in $n$. Here we derive, for a random $d$-regular graph, the limiting probability that ${cal L}(G)$ simultaneously contains the entire range ${ell,ldots,n}$ for $ellgeq 3$, as an explicit expression $theta_ell=theta_ell(d)in(0,1)$ which goes to $1$ as $elltoinfty$. For the random graph ${cal G}(n,p)$ with $p=c/n$, where $cgeq C_0$ for some absolute constant $C_0$, we show the analogous result for the range ${ell,ldots,(1-o(1))L_{max}(G)}$, where $L_{max}$ is the length of a longest cycle in $G$. The limiting probability for ${cal G}(n,p)$ coincides with $theta_ell$ from the $d$-regular case when $c$ is the integer $d-1$. In addition, for the directed random graph ${cal D}(n,p)$ we show results analogous to those on ${cal G}(n,p)$, and for both models we find an interval of $c epsilon^2 n$ consecutive cycle lengths in the slightly supercritical regime $p=frac{1+epsilon}n$." @default.
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- W3082020172 date "2020-08-31" @default.
- W3082020172 modified "2023-09-28" @default.
- W3082020172 title "Cycle lengths in sparse random graphs" @default.
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