Fractional Domatic Number in Graph Theory

• Fractional Domatic Number is the LP relaxation of the domatic number, measuring the maximum weighted packing of dominating sets under per-vertex capacity constraints.
• It exhibits a rigid threshold behavior where values exactly 1 and 2 correspond to graphs with isolated vertices and specific low-degree structures such as leaves or C4-components.
• Extremal theory links FD(G) to minimum degree, offering tight bounds and probabilistic constructions that highlight the parameter’s significance in both planar and general graphs.

Fractional domatic number is the fractional relaxation of the domatic number of a graph. Whereas the ordinary domatic number asks for the maximum number of pairwise disjoint dominating sets in a partition of the vertex set, the fractional parameter replaces disjointness by per-vertex capacity constraints and maximizes total packed weight. In the recent literature it is denoted $FD(G)$ or $\fdom(G)$, and it admits equivalent formulations via multisets of dominating sets, linear programming, random dominating sets, and dominating $(p:q)$-colorings. The parameter has a rigid threshold structure at the values $1$ and $2$, a sharp extremal theory in terms of minimum degree, and a closely related connected variant based on connected dominating sets and node capacities (Gadouleau et al., 2023, Chuet et al., 27 Aug 2025, Ene et al., 2013).

1. Definition and equivalent formulations

Let $G=(V,E)$ be a finite simple undirected graph. A set $D\subseteq V$ is a dominating set if every vertex of $V\setminus D$ has a neighbor in $D$. The domatic number is the maximum number of pairwise disjoint dominating sets whose union is $V$.

For a natural number $\fdom(G)$0, let $\fdom(G)$1 be the maximum number of dominating sets, not necessarily distinct, such that every vertex of $\fdom(G)$2 belongs to at most $\fdom(G)$3 of them. The fractional domatic number is then

$\fdom(G)$4

Equivalently, a multiset $\fdom(G)$5 of dominating sets is a $\fdom(G)$6-configuration if every vertex belongs to at most $\fdom(G)$7 members of $\fdom(G)$8, and

$\fdom(G)$9

where the maximum ranges over all natural numbers $(p:q)$0 such that $(p:q)$1 admits a $(p:q)$2-configuration. Because this is the LP relaxation of the integer program for the domatic number, the supremum is attained, so “supremum” can be replaced by “maximum” (Gadouleau et al., 2023).

If $(p:q)$3 denotes the family of all dominating sets of $(p:q)$4, the linear program studied in the later paper is

$(p:q)$5

This is the standard packing formulation: assign nonnegative weights to dominating sets so that the total weight of dominating sets containing any fixed vertex is at most $(p:q)$6 (Chuet et al., 27 Aug 2025).

The same paper emphasizes two further equivalent viewpoints. In probabilistic form,

$(p:q)$7

In coloring form, a dominating $(p:q)$8-coloring is a map

$(p:q)$9

such that every closed neighborhood sees all $1$0 colors,

$1$1

and

$1$2

The dual LP is the minimum-weight fractional bottleneck: $1$3 and by strong duality $1$4 (Chuet et al., 27 Aug 2025).

2. Threshold behavior at the values $1$5 and $1$6

The smallest possible values of the parameter are completely characterized. Graphs with fractional domatic number $1$7 are exactly the graphs that contain an isolated vertex. If $1$8 is isolated, then every dominating set must contain $1$9, so in every $2$0-configuration one has $2$1, hence $2$2; conversely, if a graph has no isolated vertices, then $2$3 (Gadouleau et al., 2023).

The lower bound $2$4 for graphs without isolated vertices is obtained from Ore’s observation that if $2$5 is a graph without isolated vertices, then the complement of a minimal dominating set of $2$6 is also a dominating set. Thus values $2$7 and $2$8 are the natural minimum thresholds: $2$9 is the absolute minimum, and once isolated vertices are absent, $G=(V,E)$0 becomes the minimum possible value (Gadouleau et al., 2023).

The value $G=(V,E)$1 is also characterized exactly. A graph without isolated vertices has fractional domatic number $G=(V,E)$2 if and only if it has a vertex of degree $G=(V,E)$3 or a connected component isomorphic to a $G=(V,E)$4-cycle. Equivalently,

$G=(V,E)$5

This yields a particularly rigid description of the landscape near the minimum (Gadouleau et al., 2023).

Several exact examples are known. For cycles,

$G=(V,E)$6

In particular, $G=(V,E)$7, $G=(V,E)$8, and $G=(V,E)$9 for every cycle $D\subseteq V$0. For disjoint unions,

$D\subseteq V$1

so a single $D\subseteq V$2-component forces the entire graph to have fractional domatic number $D\subseteq V$3 (Gadouleau et al., 2023).

3. Extremal theory in terms of minimum degree

A general extremal theorem states that if $D\subseteq V$4 has minimum degree $D\subseteq V$5, then

$D\subseteq V$6

The upper bound is obtained by taking a minimum-degree vertex $D\subseteq V$7: every dominating set intersects the closed neighborhood $D\subseteq V$8, and $D\subseteq V$9. The lower bound is probabilistic: choose each vertex independently with probability

$V\setminus D$0

form a random set $V\setminus D$1, then add every vertex not dominated by $V\setminus D$2, producing a random dominating set $V\setminus D$3. For every vertex $V\setminus D$4,

$V\setminus D$5

which yields the lower bound via the probabilistic formulation. As $V\setminus D$6, this is

$V\setminus D$7

and this is asymptotically tight even within the class of split graphs (Chuet et al., 27 Aug 2025).

The case $V\setminus D$8 is substantially more rigid. Excluding eight exceptional graphs, the fractional domatic number of every connected graph of minimum degree at least $V\setminus D$9 is at least $D$0. The exceptional family consists of $D$1, $D$2, $D$3, and five further small graphs described explicitly in the paper. Since every graph in that family except $D$4 has a dominating $D$5-coloring, it follows in particular that every connected graph $D$6 with minimum degree at least $D$7 has

$D$8

This strengthens the earlier conjecture that the next value above $D$9 should be $V$0 (Chuet et al., 27 Aug 2025).

The lower bound $V$1 is best possible in a strong sense. If a graph contains a hammock, then

$V$2

Moreover, for every integer $V$3, there exists a bipartite graph $V$4 of minimum degree $V$5 and girth $V$6 such that

$V$7

These values converge to $V$8 from above, so $V$9 cannot be improved as a universal lower bound by excluding only finitely many graphs, even when restricting to bipartite graphs of girth at least $\fdom(G)$00 (Chuet et al., 27 Aug 2025).

For planar graphs of minimum degree at least $\fdom(G)$01, large girth drives the parameter toward the natural upper bound $\fdom(G)$02. More precisely, for every $\fdom(G)$03, every planar graph $\fdom(G)$04 of minimum degree at least $\fdom(G)$05 and girth at least $\fdom(G)$06 has

$\fdom(G)$07

and therefore

$\fdom(G)$08

The proof uses long suspended paths and extension lemmas for fractional colorings (Chuet et al., 27 Aug 2025).

4. Relation to connected and capacitated variants

A distinct but closely related parameter is the fractional connected domatic number. A dominating set $\fdom(G)$09 is connected if the subgraph induced by $\fdom(G)$10 is connected. A fractional connected domatic packing assigns nonnegative weight $\fdom(G)$11 to each connected dominating set $\fdom(G)$12 such that

$\fdom(G)$13

and its optimum is the fractional connected domatic number. The node-capacitated generalization assigns each vertex $\fdom(G)$14 a nonnegative capacity $\fdom(G)$15 and requires

$\fdom(G)$16

If $\fdom(G)$17 for all $\fdom(G)$18, this becomes exactly the ordinary fractional connected domatic packing problem (Ene et al., 2013).

The controlling upper-bound parameter in the connected setting is the minimum capacity of a node separator. If

$\fdom(G)$19

then fractionally one can pack at most $\fdom(G)$20 connected dominating sets. The reason is that every connected dominating set must intersect every node separator, so each packed unit consumes at least one unit of separator capacity (Ene et al., 2013).

Algorithmically, the paper proves that there is a polynomial-time algorithm constructing a fractional connected domatic packing of size $\fdom(G)$21 for node-capacitated planar and minor-closed families of graphs, and a polynomial-time algorithm constructing a packing of size

$\fdom(G)$22

for node-capacitated general graphs. These guarantees are obtained from LP rounding for minimum-cost connected dominating set and a Carr–Vempala decomposition theorem (Ene et al., 2013).

This connected theory is adjacent to, but not identical with, the ordinary fractional domatic number. The packed objects are connected dominating sets rather than arbitrary dominating sets, and the natural upper-bound parameter is minimum separator capacity rather than minimum degree. The distinction is substantive: the 2013 paper is fundamentally about the connected variant, while the 2023 and 2025 papers treat the ordinary domination-based parameter (Ene et al., 2013).

5. Related generalizations and the $\fdom(G)$23-tuple viewpoint

A further generalization replaces domination by $\fdom(G)$24-tuple domination. For a positive integer $\fdom(G)$25, a set $\fdom(G)$26 is a $\fdom(G)$27-tuple dominating set if every vertex of $\fdom(G)$28 is adjacent to at least $\fdom(G)$29 vertices of $\fdom(G)$30 and every vertex of $\fdom(G)$31 is adjacent to at least $\fdom(G)$32 vertices in $\fdom(G)$33. Equivalently,

$\fdom(G)$34

The $\fdom(G)$35-tuple domatic number $\fdom(G)$36 is the largest number of sets in a partition of $\fdom(G)$37 into $\fdom(G)$38-tuple dominating sets; when $\fdom(G)$39, this is the ordinary domatic number (Kazemi, 2011).

The integral theory satisfies two bounds that are directly suggestive for fractional packings: $\fdom(G)$40 and

$\fdom(G)$41

The second bound follows from the fact that every $\fdom(G)$42-tuple dominating set intersects every closed neighborhood $\fdom(G)$43 in at least $\fdom(G)$44 vertices. The paper also proves a lower bound

$\fdom(G)$45

obtained from the stronger statement that every subset $\fdom(G)$46 of size at least $\fdom(G)$47 is a $\fdom(G)$48-tuple dominating set (Kazemi, 2011).

The paper does not explicitly define or analyze a fractional $\fdom(G)$49-tuple domatic number, but its partition viewpoint suggests the natural packing LP over $\fdom(G)$50-tuple dominating sets. A plausible implication is that the same counting arguments extend the upper bounds

$\fdom(G)$51

to the fractional setting. In that sense, the $\fdom(G)$52-tuple theory supplies a closed-neighborhood generalization of the standard domatic framework (Kazemi, 2011).

6. Spectrum, complexity, and open directions

The value spectrum near the minimum is unusually rigid. The 2023 paper proved that graphs with fractional domatic number $\fdom(G)$53 are exactly the graphs with an isolated vertex, and that among graphs without isolated vertices, the value $\fdom(G)$54 occurs exactly for graphs with a leaf or a $\fdom(G)$55-component. It also conjectured that if the fractional domatic number is more than $\fdom(G)$56, then it is at least $\fdom(G)$57 (Gadouleau et al., 2023).

The 2025 paper substantially sharpened that picture. It proves that, for connected graphs of minimum degree at least $\fdom(G)$58, excluding eight exceptional graphs, the value is at least $\fdom(G)$59; it also notes that there are no values in

$\fdom(G)$60

but there are infinitely many values accumulating above $\fdom(G)$61. Several open problems remain: characterize graphs with $\fdom(G)$62; describe all possible values of $\fdom(G)$63; determine whether the large-girth planar phenomenon has a nonplanar analogue; and locate a sharp complexity threshold for decision problems of the form $\fdom(G)$64. The same paper proves that deciding whether $\fdom(G)$65 is NP-hard, even for split graphs (Chuet et al., 27 Aug 2025).

A persistent source of confusion is the relation among three different parameters: ordinary domatic number, fractional domatic number, and fractional connected domatic number. The first is an integral partition parameter, the second is its LP relaxation over dominating sets, and the third is the analogous LP relaxation over connected dominating sets. Another common misconception is that minimum degree alone governs the parameter exactly; the minimum-degree theory is powerful, but the exact threshold behavior at small values depends on finer obstructions such as isolated vertices, leaves, $\fdom(G)$66-components, and, in the connected setting, node separators (Gadouleau et al., 2023, Chuet et al., 27 Aug 2025).

Overall, the modern theory presents fractional domatic number as a well-structured packing parameter: it has clean LP and probabilistic formulations, exact threshold characterizations at the lowest values, asymptotically tight degree-based bounds, a sharp low-degree theory, and a distinct connected counterpart governed by separator capacity rather than minimum degree.