Fair Coalition Number in Graph Theory

• Fair Coalition Number is a domination-based invariant defined by partitioning a graph into sets where non-dominating parts form pairs that together provide exact k-neighbor coverage.
• The parameter is developed through constructive fair coalition partitions and is bounded using the fair domination number and fair domatic partitions.
• Exact values and periodic bounds for families such as paths, cycles, coronas, and cubic graphs illustrate its applications and structural implications in domination theory.

The fair coalition number $\mathcal{C}_f(G)$ is a domination-theoretic partition parameter defined for a simple graph $G=(V,E)$ by replacing ordinary domination with fair domination. A dominating set $D\subseteq V$ is a $k$-fair dominating set if $|N(v)\cap D|=k$ for every vertex $v\in V\setminus D$, for some integer $k\ge 1$; a fair dominating set is any $k$-fair dominating set. A fair coalition is a pair of disjoint subsets $A_1,A_2\subseteq V$ such that neither $A_1$ nor $G=(V,E)$0 is a fair dominating set of $G=(V,E)$1, while $G=(V,E)$2 is a fair dominating set. A fair coalition partition is a partition of $G=(V,E)$3 in which every part is either a singleton fair dominating set or a non-fair dominating set that forms a fair coalition with another non-fair dominating part. The fair coalition number is the maximum cardinality of a fair coalition partition, and is denoted $G=(V,E)$4 (Alikhani et al., 20 Jul 2025).

1. Formal framework

The underlying notion is fair domination. For a graph $G=(V,E)$5, a dominating set $G=(V,E)$6 is called a $G=(V,E)$7-fair dominating set ($G=(V,E)$8-set) if

$G=(V,E)$9

for some integer $D\subseteq V$0. The paper introducing fair coalitions also uses $D\subseteq V$1 for the minimum cardinality of an $D\subseteq V$2-set and $D\subseteq V$3 for the minimum cardinality among all fair dominating sets. It states the inequalities

$D\subseteq V$4

and $D\subseteq V$5 if and only if $D\subseteq V$6 (Alikhani et al., 20 Jul 2025).

A fair coalition in $D\subseteq V$7 consists of two disjoint sets $D\subseteq V$8 and $D\subseteq V$9 such that neither set is a fair dominating set of $k$0, but $k$1 is a fair dominating set. A fair coalition partition, abbreviated $k$2-partition, is a partition

$k$3

of $k$4 such that every part $k$5 is either a singleton fair dominating set of $k$6, or is not a fair dominating set but forms a fair coalition with another non-fair dominating set $k$7. The fair coalition number $k$8 is the maximum cardinality of such a partition; a partition attaining this maximum is called a $k$9-partition (Alikhani et al., 20 Jul 2025).

The parameter therefore combines two ingredients: the exact-neighborhood requirement of fair domination and the coalition-partition requirement that non-fair parts become admissible only through cooperation. In the terminology of the source paper, it measures how finely the vertex set can be partitioned while preserving this coalitional fair-domination property (Alikhani et al., 20 Jul 2025).

2. Relation to ordinary coalition numbers

The fair coalition number belongs to the broader coalition-partition literature in domination theory. In the ordinary setting, a coalition in a graph $|N(v)\cap D|=k$0 is a pair of disjoint sets $|N(v)\cap D|=k$1 such that neither $|N(v)\cap D|=k$2 nor $|N(v)\cap D|=k$3 is a dominating set, but $|N(v)\cap D|=k$4 is a dominating set. A coalition partition is a partition of $|N(v)\cap D|=k$5 in which each part is either a dominating set with cardinality $|N(v)\cap D|=k$6, or is not a dominating set but forms a coalition with another part; the maximum cardinality of such a partition is the coalition number, denoted $|N(v)\cap D|=k$7 or $|N(v)\cap D|=k$8 in the literature (Bakhshesh et al., 2021).

This distinction is substantive rather than terminological. The 2024 paper on cubic graphs explicitly notes that it does not use the terminology “fair coalition number” and instead studies the standard coalition number $|N(v)\cap D|=k$9, where the relevant union condition is ordinary domination rather than fair domination (Dobrynin et al., 2024). The fair coalition number is thus not a renaming of $v\in V\setminus D$0; it is a separate invariant obtained by replacing domination by fair domination.

The same literature also contains a $v\in V\setminus D$1-domination analogue. In a $v\in V\setminus D$2-coalition, the union of the two parts must be a $v\in V\setminus D$3-dominating set, meaning every vertex outside the union has at least $v\in V\setminus D$4 neighbors in it, and the associated maximum partition size is the $v\in V\setminus D$5-coalition number $v\in V\setminus D$6 (Jafari et al., 2024). By comparison, fair coalitions impose exact counts $v\in V\setminus D$7, not lower bounds. This places $v\in V\setminus D$8 in a more restrictive branch of the coalition framework.

3. Bounds and structural principles

The initial theory of $v\in V\setminus D$9 connects it to fair domatic partitions. A fair domatic partition is a partition of $k\ge 1$0 into fair dominating sets, and its maximum size is the fair domatic number $k\ge 1$1. A key theorem states that if $k\ge 1$2 is a graph of order $k\ge 1$3 without full vertices, then

$k\ge 1$4

The proof idea given in the source is that each fair dominating set in a fair domatic partition can be split into two non-fair sets that together form a fair coalition, yielding a fair coalition partition with at least twice as many parts (Alikhani et al., 20 Jul 2025).

The main upper bounds are expressed in terms of the fair domination number. If $k\ge 1$5 has order $k\ge 1$6 and fair domination number $k\ge 1$7, then

$k\ge 1$8

If $k\ge 1$9 is connected with order $k$0 and fair domination number $k$1, then

$k$2

The source explains these bounds by observing that, in a $k$3-partition, at least one pair of parts must union to a fair dominating set, so their total size is at least $k$4 (Alikhani et al., 20 Jul 2025).

A sharp special case occurs for coronas of trees. If $k$5 is a tree of order $k$6 of the form $k$7, then

$k$8

and the paper proves the exact value

$k$9

The argument uses the fact that the set of leaves of $A_1,A_2\subseteq V$0 forms a $A_1,A_2\subseteq V$1-FD-set and that the fair domatic number of such a tree is $A_1,A_2\subseteq V$2 (Alikhani et al., 20 Jul 2025).

4. Exact values for paths, cycles, and corona trees

The source paper determines $A_1,A_2\subseteq V$3 exactly for several standard graph families. For paths,

$A_1,A_2\subseteq V$4

For cycles, the value depends on the residue class of $A_1,A_2\subseteq V$5 modulo $A_1,A_2\subseteq V$6: $A_1,A_2\subseteq V$7

$A_1,A_2\subseteq V$8

$A_1,A_2\subseteq V$9

For trees of the form $A_1$0, the exact value is

$A_1$1

when $A_1$2 (Alikhani et al., 20 Jul 2025).

Graph family	Exact value of $A_1$3
$A_1$4, $A_1$5	$A_1$6
$A_1$7, $A_1$8	$A_1$9
$G=(V,E)$00, $G=(V,E)$01	$G=(V,E)$02
$G=(V,E)$03, $G=(V,E)$04	$G=(V,E)$05
$G=(V,E)$06, $G=(V,E)$07	$G=(V,E)$08

The proofs are constructive. For even paths $G=(V,E)$09, the paper gives a four-part partition

$G=(V,E)$10

with

$G=(V,E)$11

and states that $G=(V,E)$12 and $G=(V,E)$13 are partners. For odd paths $G=(V,E)$14, it also gives a four-part construction, now with $G=(V,E)$15 having several partner relations. For cycles, the constructions similarly depend on congruence classes: six-part partitions for $G=(V,E)$16, five-part partitions for $G=(V,E)$17, and four-part partitions for $G=(V,E)$18, together with explicitly stated partner pairs (Alikhani et al., 20 Jul 2025).

These formulas show that fair coalitional structure behaves differently from ordinary coalition number on the same families. In particular, paths have constant fair coalition number $G=(V,E)$19 for all $G=(V,E)$20, while cycles exhibit a periodic dependence on $G=(V,E)$21 (Alikhani et al., 20 Jul 2025).

5. Cubic graphs of small order

The paper also computes $G=(V,E)$22 for the cubic graphs of orders $G=(V,E)$23, $G=(V,E)$24, and $G=(V,E)$25 that it studies. For order $G=(V,E)$26, there are exactly two cubic graphs, denoted $G=(V,E)$27 and $G=(V,E)$28, and both satisfy

$G=(V,E)$29

The witnessing $G=(V,E)$30-partition is the partition into all singleton sets (Alikhani et al., 20 Jul 2025).

For order $G=(V,E)$31, there are six cubic graphs $G=(V,E)$32, with values

$G=(V,E)$33

For $G=(V,E)$34, $G=(V,E)$35, and $G=(V,E)$36, the partition into $G=(V,E)$37 singletons works. For $G=(V,E)$38 and $G=(V,E)$39, the paper gives maximum partitions containing both singleton and two-vertex parts (Alikhani et al., 20 Jul 2025).

For order $G=(V,E)$40, the $G=(V,E)$41 cubic graphs $G=(V,E)$42 are partitioned into three value classes:

Order	Value of $G=(V,E)$43	Graphs
$G=(V,E)$44	$G=(V,E)$45	$G=(V,E)$46
$G=(V,E)$47	$G=(V,E)$48	$G=(V,E)$49
$G=(V,E)$50	$G=(V,E)$51	$G=(V,E)$52
$G=(V,E)$53	$G=(V,E)$54	$G=(V,E)$55
$G=(V,E)$56	$G=(V,E)$57	$G=(V,E)$58
$G=(V,E)$59	$G=(V,E)$60	$G=(V,E)$61
$G=(V,E)$62	$G=(V,E)$63	$G=(V,E)$64

A notable case is the Petersen graph $G=(V,E)$65, which is $G=(V,E)$66 in the paper’s labeling and satisfies

$G=(V,E)$67

The computations are again based on explicit $G=(V,E)$68-partitions and partner relations. In some cases, all singleton partitions are feasible; in others, maximum partitions require larger blocks, reflecting the constraints imposed by fair domination (Alikhani et al., 20 Jul 2025).

These small-order cubic computations are distinct from the ordinary coalition-number results for cubic graphs. In the standard domination-based theory, earlier work computed coalition numbers for cubic graphs of order at most $G=(V,E)$69, and later work constructed an infinite family of cubic graphs with maximal ordinary coalition number $G=(V,E)$70 (Alikhani et al., 2022).

6. Generalizations, variants, and terminological boundaries

A direct extension of the fair coalition number is the $G=(V,E)$71-fair coalition number $G=(V,E)$72. In that setting, a set $G=(V,E)$73 is a $G=(V,E)$74-fair dominating set if every vertex not in $G=(V,E)$75 has exactly $G=(V,E)$76 neighbors in $G=(V,E)$77; a $G=(V,E)$78-fair coalition is a pair of disjoint sets whose union is a $G=(V,E)$79-fair dominating set while neither set is $G=(V,E)$80-fair dominating individually; and a $G=(V,E)$81-fair coalition partition is a partition in which each part is either a $G=(V,E)$82-fair dominating set with exactly $G=(V,E)$83 vertices or forms a $G=(V,E)$84-fair coalition with another part. The $G=(V,E)$85-fair coalition number is the maximum size of such a partition (Jafari et al., 14 Sep 2025).

The 2025 $G=(V,E)$86-fair paper establishes several general bounds: $G=(V,E)$87

$G=(V,E)$88

and, for $G=(V,E)$89-regular graphs,

$G=(V,E)$90

It also gives exact values for several families, including

$G=(V,E)$91

$G=(V,E)$92

together with tree bounds such as

$G=(V,E)$93

from which it derives that $G=(V,E)$94 only for $G=(V,E)$95 and $G=(V,E)$96 only for $G=(V,E)$97 or $G=(V,E)$98 (Jafari et al., 14 Sep 2025).

Several adjacent notions should not be conflated with fair coalition number. The $G=(V,E)$99-coalition number $D\subseteq V$00 concerns $D\subseteq V$01-domination rather than $D\subseteq V$02-fair domination (Jafari et al., 2024). The restrained coalition number $D\subseteq V$03 replaces domination by restrained domination and satisfies

$D\subseteq V$04

for every graph (Dobrynin et al., 12 Dec 2025). Finally, a 2026 paper on distributed coalition-value calculations in characteristic function games uses the language of fairness for allocation balance, but explicitly does not define a standalone term “Fair Coalition Number”; instead it studies equitable allocation and balanced load with tight bounds on the number of coalitions assigned to each agent (Payne et al., 18 Apr 2026).

Within graph theory proper, the fair coalition number $D\subseteq V$05 is therefore best understood as a domination-based partition invariant rooted in exact neighborhood counts. Its current theory is built from explicit constructions, domination-number bounds, and exact evaluations on graph families such as paths, cycles, coronas, and small cubic graphs (Alikhani et al., 20 Jul 2025).