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arXiv · 2611.00437

Two-point concentration for graph classes admitting sparse vertex deletion

Margarita Akhmejanova·Vladislav Kozhevnikov
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Abstract

For fixed $p\in(0,1)$, we prove that the maximum number of vertices in an induced subgraph of $G(n,p)$ from a given class is concentrated on two consecutive integers, unifying and extending several earlier results. We require that the class contain arbitrarily large graphs and that every sufficiently large $k$-vertex member have a vertex of degree $o(k)$ whose deletion leaves a graph in the same class. This condition holds for empty graphs, trees, forests, planar graphs, graphs of bounded maximum degree, degeneracy or treewidth, graphs excluding a fixed complete bipartite subgraph (including graphs containing no cycle of length four), and many other graph classes.

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Margarita Akhmejanova, Vladislav Kozhevnikov. 2026-10-30. Two-point concentration for graph classes admitting sparse vertex deletion. https://arxiv.org/abs/2611.00437

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