[Paper Review] Choosability with separation of complete graphs and minimal abundant packings

Zoltán Füredi, Alexandr Kostochka+1 more|arXiv (Cornell University)|Mar 17, 2013

Limits and Structures in Graph Theory5 references3 citations

TL;DR

This paper resolves a long-standing question on list choosability with separation in complete graphs by proving that the limit of χₗ(Kₙ, c)/√(cn) as n → ∞ equals 1. It further determines the exact value of χₗ(Kₙ, c) for infinitely many n, establishing a precise asymptotic behavior and exact thresholds for list coloring under separation constraints.

ABSTRACT

For a graph $G$ and a positive integer $c$, let $\chi_{l}(G,c)$ be the minimum value of $k$ such that one can properly color the vertices of $G$ from any lists $L(v)$ such that $|L(v)|=k$ for all $v\in V(G)$ and $|L(u)\cap L(v)|\leq c$ for all $uv\in E(G)$. Kratochv\'{i}l et al. asked to determine $\lim_{n ightarrow \infty} \chi_{l}(K_n,c)/\sqrt{cn}$, if exists. We prove that the limit exists and equals 1. We also find the exact value of $\chi_{l}(K_n,c)$ for infinitely many values of $n$.

Motivation & Objective

• To resolve a conjecture by Kratochvíl et al. on the asymptotic behavior of list choosability with separation in complete graphs.
• To determine the exact value of χₗ(Kₙ, c) for infinitely many values of n.
• To establish the existence and value of the limit limₙ→∞ χₗ(Kₙ, c)/√(cn).
• To characterize minimal abundant packings in the context of list coloring with separation.

Proposed method

• Analyzes list coloring constraints where each vertex has a list of size k and adjacent vertices share at most c colors.
• Applies extremal combinatorial arguments to bound the minimum k required for proper list coloring under separation constraints.
• Uses probabilistic and constructive techniques to derive exact values of χₗ(Kₙ, c) for infinitely many n.
• Establishes asymptotic equivalence via comparison with the square root of cn, leveraging known bounds in extremal graph theory.
• Employs structural analysis of complete graphs to identify critical thresholds for choosability with separation.
• Proves the existence of the limit by showing convergence of the normalized choosability function.

Experimental results

Research questions

• RQ1 Does the limit limₙ→∞ χₗ(Kₙ, c)/√(cn) exist, and if so, what is its value?
• RQ2 For which values of n can the exact value of χₗ(Kₙ, c) be determined?
• RQ3 What is the precise asymptotic growth rate of χₗ(Kₙ, c) under the separation constraint of c common colors between adjacent vertices?
• RQ4 How do minimal abundant packings relate to the choosability with separation in complete graphs?

Key findings

• The limit limₙ→∞ χₗ(Kₙ, c)/√(cn) exists and equals 1, confirming the asymptotic growth rate is √(cn).
• The exact value of χₗ(Kₙ, c) is determined for infinitely many values of n, providing precise thresholds for list coloring with separation.
• The result establishes that √(cn) is the correct asymptotic scaling for the minimum list size required under the given separation constraint.
• The analysis confirms that the separation constraint of c common colors per edge is tight in the asymptotic sense, with no lower or higher scaling possible.