Fractional domatic number and minimum degree

2025-08-27

AI Summary

This paper investigates the extremal problem of the fractional directed cover number in graphs with minimum degree $d$, focusing on the tight lower bound for minimum degree $delta = 2$. Employing a combination of linear programming relaxation and extremal graph theory, we prove that, except for finitely many exceptional graphs, every connected graph with minimum degree at least $2$ has fractional directed cover number at least $5/2$, thereby fully confirming and strengthening a conjecture by Gadouleau et al.; this bound is asymptotically tight and cannot be improved when exceptions are excluded. Furthermore, we establish an asymptotically tight bound for general minimum degree $d$, and show that the fractional directed cover number of large-girth planar graphs approaches $3$. Our results unify the structural behavior of split graphs, bipartite graphs, and planar graphs, offering a new paradigm for extremal analysis of fractional domination-type parameters.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Planning under Uncertainty
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Social Networks and Social Media: Applications and emergent phenomena

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 $d o infty$. 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 $g oinfty$.

Problem

• Studying fractional domatic number bounds in graphs with minimum degree d
• Determining extremal values for fractional domatic number in degree-constrained graphs
• Establishing improved lower bounds for connected graphs with minimum degree 2

Innovation

• LP-relaxation of domatic number
• Minimum degree graphs analysis
• Planar graphs girth bounds