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Mathematics > Combinatorics

arXiv:2302.11668 (math)
[Submitted on 22 Feb 2023 (v1), last revised 13 Oct 2023 (this version, v2)]

Title:Graphs with minimum fractional domatic number

Authors:Maximilien Gadouleau, Nathaniel Harms, George B. Mertzios, Viktor Zamaraev
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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.
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
Cite as: arXiv:2302.11668 [math.CO]
  (or arXiv:2302.11668v2 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.2302.11668

Submission history

From: Viktor Zamaraev [view email]
[v1] Wed, 22 Feb 2023 21:44:30 UTC (14 KB)
[v2] Fri, 13 Oct 2023 18:35:06 UTC (13 KB)
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