On Coupon Colorings of Graphs

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 $χ_{c}(G)$. In this paper, we prove that every $d$-regular graph $G$ has $χ_{c}(G) \geq (1 - o(1))d/\log d$ as $d \rightarrow \infty$, and the proportion of $d$-regular graphs $G$ for which $χ_c(G) \leq (1 + o(1))d/\log d$ tends to $1$ as $|V(G)| \rightarrow \infty$.

No reviews yet.

No comments yet.