Long paths and cycles in random subgraphs of \(\mathcal{H}\)-free graphs

Michael Krivelevich¹ ; Wojciech Samotij¹

¹Tel Aviv University

The electronic journal of combinatorics, Tome 21 (2014) no. 1

Résumé

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive $\varepsilon$, there exists a positive $\delta$ (depending only on $\varepsilon$) such that the following holds: If $p \geq \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(\delta k)$, where $n_\mathcal{H}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.

Détail

DOI : 10.37236/3198

Classification : 05C80, 05C38, 05C12, 05C40
Mots-clés : random graphs, forbidden subgraphs