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Communication Dans Un Congrès Année : 2022

## On List Coloring With Separation Of The Complete Graph And Set System Intersections

Olivier Togni
Jean-Christophe Godin
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#### Résumé

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 Poincaré'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.

### Dates et versions

hal-04001196 , version 1 (22-02-2023)

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• HAL Id : hal-04001196 , version 1

### Citer

Olivier Togni, Jean-Christophe Godin, Rémi Grisot. On List Coloring With Separation Of The Complete Graph And Set System Intersections. The 11th International Colloquium on Graph Theory and combinatorics, Jul 2022, Montpellier, France. ⟨hal-04001196⟩

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