On Coupon Colorings of Graphs

TL;DR

This paper investigates the coupon coloring number of graphs, establishing asymptotic bounds for $d$-regular graphs and showing most such graphs have a coupon coloring number close to $d/ ext{log} d$ as the number of vertices grows.

Contribution

It proves that the coupon coloring number of large $d$-regular graphs is asymptotically at least $(1 - o(1))d/ ext{log} d$ and that most such graphs have this number close to this bound.

Findings

$oxed{ ext{Coupon coloring number} ext{ of } d ext{-regular graphs} ext{ is at least } (1 - o(1))d/ ext{log} d$.

Most large $d$-regular graphs have a coupon coloring number at most $(1 + o(1))d/ ext{log} d$.

Abstract

Let $G$ be a graph with no isolated vertices. A {\em $k$-coupon coloring} of $G$ is an assignment of colors from $[k] := \{1,2,\dots,k\}$ to the vertices of $G$ such that the neighborhood of every vertex of $G$ contains vertices of all colors from $[k]$. The maximum $k$ for which a $k$-coupon coloring exists is called the {\em coupon coloring number} of $G$, and is denoted $\chi_{c}(G)$. In this paper, we prove that every $d$-regular graph $G$ has $\chi_{c}(G) \geq (1 - o(1))d/\log d$ as $d \rightarrow \infty$, and the proportion of $d$-regular graphs $G$ for which $\chi_c(G) \leq (1 + o(1))d/\log d$ tends to $1$ as $|V(G)| \rightarrow \infty$.