Graphs with minimum fractional domatic number

Gadouleau, Maximilien; Harms, Nathaniel; Mertzios, George B.; Zamaraev, Viktor

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.

Citation

Gadouleau, M., Harms, N., Mertzios, G. B., & Zamaraev, V. (2024). Graphs with minimum fractional domatic number. Discrete Applied Mathematics, 343, 140-148. https://doi.org/10.1016/j.dam.2023.10.020

• Journal Article Type: Article
• Acceptance Date: Oct 12, 2023
• Online Publication Date: Oct 28, 2023
• Publication Date: Jan 30, 2024
• Deposit Date: Nov 6, 2023
• Publicly Available Date: Nov 7, 2023
• Journal: Discrete Applied Mathematics
• Print ISSN: 0166-218X
• Electronic ISSN: 1872-6771
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 343
• Pages: 140-148
• DOI: https://doi.org/10.1016/j.dam.2023.10.020
• Keywords: Applied Mathematics; Discrete Mathematics and Combinatorics
• Public URL: https://durham-repository.worktribe.com/output/1897822