Fractional domatic number and minimum degree
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 $g\to\infty$.