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arXiv · 2107.06977

On subgraphs with degrees of prescribed residues in the random graph

Asaf Ferber·Liam Hardiman·Michael Krivelevich
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Abstract

We show that with high probability the random graph $G_{n, 1/2}$ has an induced subgraph of linear size, all of whose degrees are congruent to $r\pmod q$ for any fixed $r$ and $q\geq 2$. More generally, the same is true for any fixed distribution of degrees modulo $q$. Finally, we show that with high probability we can partition the vertices of $G_{n, 1/2}$ into $q+1$ parts of nearly equal size, each of which induces a subgraph all of whose degrees are congruent to $r\pmod q$. Our results resolve affirmatively a conjecture of Scott, who addressed the case $q=2$.

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Asaf Ferber, Liam Hardiman, Michael Krivelevich. 2021-07-14. On subgraphs with degrees of prescribed residues in the random graph. https://arxiv.org/abs/2107.06977

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