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

TL;DR

This paper investigates the maximum intersection size in list coloring with separation for complete graphs, providing exact values for some parameters and bounds for others using advanced combinatorial techniques.

Contribution

It introduces an improved method based on set partitions to determine the separation number for complete graphs in list coloring with separation.

Findings

Exact separation numbers for certain parameters of complete graphs

Bounds established for remaining parameter ranges

Enhanced combinatorial techniques for list coloring problems

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.