# Graphs with minimum fractional domatic number

## TL;DR

This paper characterizes graphs with fractional domatic number exactly 2, showing they must contain a degree-1 vertex or a 4-cycle component, and conjectures a lower bound for higher fractional domatic numbers.

## Contribution

It provides a complete characterization of graphs with fractional domatic number 2 and proposes a conjecture for the minimal fractional domatic number exceeding 2.

## Key findings

- Graphs with fractional domatic number 1 have an isolated vertex.
- Graphs with fractional domatic number 2 contain a degree-1 vertex or a 4-cycle component.
- Conjecture: fractional domatic number greater than 2 is at least 7/3.

## Abstract

The domatic number of a graph is the maximum number of vertex disjoint dominating sets that partition the vertex set of the graph. In this paper we consider the fractional variant of this notion. Graphs with fractional domatic number 1 are exactly the graphs that contain an isolated vertex. Furthermore, it is known that all other graphs have fractional domatic number at least 2. In this note we characterize graphs with fractional domatic number 2. More specifically, we show that a graph without isolated vertices has fractional domatic number 2 if and only if it has a vertex of degree 1 or a connected component isomorphic to a 4-cycle. We conjecture that if the fractional domatic number is more than 2, then it is at least 7/3.