# A Note on Distance-Fall Colorings

## TL;DR

This paper introduces a new type of graph coloring called distance-$k$ fall coloring, proves its existence for certain classes of graphs, and confirms a conjecture about the size of distance-$d$-dominating sets in trees.

## Contribution

It establishes the existence of distance-$k$ fall colorings for connected 3-colorable graphs and trees, confirming an old conjecture about dominating sets in trees.

## Key findings

- Connected 3-colorable graphs have a distance-2 fall 3-coloring.
- Trees of order at least $k$ have a $k$-coloring with vertices within distance $k-1$ of every color.
- Proves the conjecture that trees have small independent distance-$d$-dominating sets.

## Abstract

We say a proper coloring of a graph is distance-$k$ fall if every vertex is within distance $k$ of at least one vertex of every color. We show that if $G$ is a connected graph of order at least $3$ that is $3$-colorable, thenit has a distance-2 fall 3-coloring. Further, for every integer $k\ge 2$, if $T$ is a tree of order at least $k$, then $T$ has a $k$-coloring such that every vertex is within distance $k-1$ of every color. This proves an old conjecture of Beineke and Henning that every tree of order $n$ has an independent distance-$d$-dominating set of size at most $n/(d + 1)$.