We consider vertex percolation on pseudo‐random d$$ d $$‐regular graphs. The previous study by the second author established the existence of phase transition from small components to a linear (in nd$$ \frac{n}{d} $$) sized component, at p=1d$$ p=\frac{1}{d} $$. In the supercritical regime, our main result recovers the sharp asymptotic of the size of the largest component, and shows that all other components are typically much smaller. Furthermore, we consider other typical properties of the largest component such as the number of edges, existence of a long cycle and expansion. In the subcritical regime, we strengthen the upper bound on the likely component size.

We consider the performance of the Depth First Search (DFS) algorithm on the random graph G n, 1+ǫ n , ǫ > 0 a small constant. Recently, Enriquez, Faraud and Ménard [2] proved that the stack U of the DFS follows a specific scaling limit, reaching the maximal height of (1 + o ǫ (1)) ǫ 2 n. Here we provide a simple analysis for the typical length of a maximum path discovered by the DFS.

We consider the performance of the Depth First Search (DFS) algorithm on the random graph $G\left(n,\frac{1+\epsilon}{n}\right)$, $\epsilon>0$ a small constant. Recently, Enriquez, Faraud and Ménard proved that the stack $U$ of the DFS follows a specific scaling limit, reaching the maximal height of $\left(1+o_{\epsilon}(1)\right)\epsilon^2n$. Here we provide a simple analysis for the typical length of a maximum path discovered by the DFS.

We consider vertex percolation on pseudo-random d−regular graphs. The previous study by the second author established the existence of phase transition from small components to a linear (in n d ) sized component, at p = 1 d . In the supercritical regime, our main result recovers the sharp asymptotic of the size of the largest component, and shows that all other components are typically much smaller. Furthermore, we consider other typical properties of the largest component such as the number of edges, existence of a long cycle and expansion. In the subcritical regime, we strengthen the upper bound on the likely component size.

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