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arXiv · 2209.03436

On List Coloring with Separation of the Complete Graph and Set System Intersections

Jean-Christophe Godin·Rémi Grisot·Olivier Togni
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Abstract

We consider the following list coloring with separation problem: Given a graph $G$ and integers $a,b$, find the largest integer $c$ such that for any list assignment $L$ of $G$ with $|L(v)|= a$ for any vertex $v$ and $|L(u)\cap L(v)|\le c$ for any edge $uv$ of $G$, there exists an assignment $\varphi$ of sets of integers to the vertices of $G$ such that $\varphi(u)\subset L(u)$ and $|\varphi(v)|=b$ for any vertex $u$ and $\varphi(u)\cap \varphi(v)=\emptyset$ for any edge $uv$. Such a value of $c$ is called the separation number of $(G,a,b)$. Using a special partition of a set of lists for which we obtain an improved version of Poincar\'e's crible, we determine the separation number of the complete graph $K_n$ for some values of $a,b$ and $n$, and prove bounds for the remaining values.

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Jean-Christophe Godin, Rémi Grisot, Olivier Togni. 2022-09-07. On List Coloring with Separation of the Complete Graph and Set System Intersections. https://arxiv.org/abs/2209.03436

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