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Note on induced paths in sparse random graphs

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arxiv 2102.09289 v1 pith:T3M24GT5 submitted 2021-02-18 math.CO

Stefan Glock This is my paper · ORCID
classification math.CO
keywords inducedepsilonrandomresultalonganalogousanswerscontains
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abstract

We show that for $d\ge d_0(\epsilon)$, with high probability, the random graph $G(n,d/n)$ contains an induced path of length $(3/2-\epsilon)\frac{n}{d}\log d$. This improves a result obtained independently by Luczak and Suen in the early 90s, and answers a question of Fernandez de la Vega. Along the way, we generalize a recent result of Cooley, Dragani\'c, Kang and Sudakov who studied the analogous problem for induced matchings.

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  1. Concentration of the maximum size of an induced subtree in moderately sparse random graphs

    math.CO 2025-06 conditional novelty 6.0 of 10

    For p = n^{-(e-2)/(3e-2)+ε}, the maximum induced tree size in G(n,p) is concentrated at two adjacent values.

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