Fractional domatic number and minimum degree

TL;DR

This paper investigates the fractional domatic number in graphs with various minimum degree conditions, establishing bounds and extremal values, and confirming a recent conjecture with implications for planar graphs.

Contribution

It provides tight bounds on the fractional domatic number for graphs with minimum degree d, especially for d=2, and proves a conjecture regarding bipartite graphs with girth at least 6.

Findings

Fractional domatic number is at most d+1 for graphs with minimum degree d.

For large d, the fractional domatic number is at least (1-o(1)) d/ln d.

Except for 8 specific graphs, connected graphs with minimum degree 2 have fractional domatic number at least 5/2.

Abstract

The domatic number of a graph $G$ is the maximum number of pairwise disjoint dominating sets of $G$. We are interested in the LP-relaxation of this parameter, which is called the fractional domatic number of $G$. We study its extremal value in the class of graphs of minimum degree $d$. The fractional domatic number of a graph of minimum degree $d$ is always at most $d+1$, and at least $(1-o(1))\, d/\ln d$ as $d\to \infty$. This is asymptotically tight even within the class of split graphs. Our main result concerns the case $d=2$; we show that, excluding $8$ exceptional graphs, the fractional domatic number of every connected graph of minimum degree (at least) $2$ is at least $5/2$. We also show that this bound cannot be improved if only finitely many graphs are excluded, even when restricting to bipartite graphs of girth at least $6$. This proves in a stronger sense a conjecture by Gadouleau, Harms, Mertzios, and Zamaraev (2024). This also extends and generalises results from McCuaig and Shepherd (1989), from Fujita, Kameda, and Yamashita (2000), and from Abbas, Egerstedt, Liu, Thomas, and Whalen (2016). Finally, we show that planar graphs of minimum degree at least $2$ and girth at least $g$ have fractional domatic number at least $3 - O(1/g)$ as $g\to\infty$.