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Colouring powers of cycles from random lists

Michael Krivelevich·Asaf Nachmias
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Abstract

Let $C_n^k$ be the $k$-th power of a cycle on $n$ vertices (i.e. the vertices of $C_n^k$ are those of the $n$-cycle, and two vertices are connected by an edge if their distance along the cycle is at most $k$). For each vertex draw uniformly at random a subset of size $c$ from a base set $S$ of size $s=s(n)$. In this paper we solve the problem of determining the asymptotic probability of the existence of a proper colouring from the lists for all fixed values of $c,k$, and growing $n$.

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Michael Krivelevich, Asaf Nachmias. 2005-12-01. Colouring powers of cycles from random lists. https://arxiv.org/abs/math/0512004

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