FAT Chromatic Number in Graph Colorings

• FAT chromatic number is a graph invariant defined via fair and tolerant colorings that balance same-color tolerance with equitable cross-class neighbor distribution.
• It incorporates precise bounds such as degree, order, and volume divisibility, and leverages spectral criteria to relate coloring parameters to graph eigenvalues.
• This concept bridges classical chromatic theory with equitable partitioning, opening research avenues in algorithmic complexity and spectral graph analysis.

The FAT chromatic number, denoted $\chi^{\mathrm{FAT}}(G)$, is a graph invariant originating from a relaxation of classical proper coloring, incorporating the dual principles of fairness and tolerance in color assignments. Introduced by Beers and Mulas, Fair and Tolerant (FAT) colorings generalize equitable and proper colorings by allowing each vertex to tolerate a prescribed fraction of same-colored neighbors while distributing the remaining neighbors evenly among other color classes. This non-proper coloration paradigm establishes a new maximal coloring parameter distinct from the chromatic number and sheds light on the interplay between local regularity and global colorability constraints (Beers et al., 21 Oct 2025, Shaebani, 18 Nov 2025).

1. Formal Definitions and Fundamental Properties

Let $G=(V,E)$ be a finite, simple, undirected graph. The standard notation for the degree of a vertex $v$ is $\deg(v)$ and $e(v,S) = |N(v)\cap S|$ for $S\subseteq V$, where $N(v)$ is the neighborhood of $v$.

FAT $k$-Coloring

A FAT $k$-coloring is a coloring %%%%10%%%% that partitions $V$ into nonempty color-classes $(V_1,\dots,V_k)$ such that for real parameters $\alpha,\beta\in[0,1]$, every $v\in V$ and $1\leq i\leq k$:

$$e(v,V_i) = \begin{cases} \alpha \cdot \deg(v), & v\notin V_i \ \beta \cdot \deg(v), & v\in V_i \end{cases}$$

with the normalization constraint $\beta + (k-1)\alpha = 1$.

• Fairness: Each vertex distributes a fraction $\alpha$ of its neighbors to each other color class.
• Tolerance: A fraction $\beta$ of a vertex's neighbors may share its color.

When $\beta=0$ (and $\alpha=1/(k-1)$), the coloring is a proper and equitable coloring.

FAT Chromatic Number

The FAT chromatic number, $\chi^{\mathrm{FAT}}(G)$, is defined as

$$\chi^{\mathrm{FAT}}(G) = \max\{k : \text{G admits a FAT } k\text{-coloring}\}$$

This parameter is always at least $1$ due to the trivial coloring, and is at most $|V|$ (as each class must be nonempty).

2. Bounds and Structural Constraints

Numerous general and structural bounds on $\chi^{\mathrm{FAT}}(G)$ have been established:

• Degree Bound: If $\delta = \min_{v\in V}\deg(v)$, then

$\chi^{\mathrm{FAT}}(G) \leq \delta+1$

This is tight for $K_n$ [(Beers et al., 21 Oct 2025), Prop. 2.1].

• Order Bound: $\chi^{\mathrm{FAT}}(G)\leq |V|$, with equality if and only if $G=K_{|V|}$.
• Volume Divisibility: In any FAT $k$-coloring with $\alpha>0$, all color classes have the same total degree volume:

$\mathrm{vol}(V_i)=2|E|/k$

Thus, $k$ divides $2|E|$ [(Beers et al., 21 Oct 2025), Prop. 2.8].

• Regular Graphs: For connected $d$-regular $G$, each color class has $|V|/k$ vertices and $k$ divides $|V|$ [(Beers et al., 21 Oct 2025), Thm. 2.5].
• Relation to Chromatic Number: Every proper $k$-coloring is a FAT coloring with $\alpha=1/(k-1)$, so

$\chi^{\mathrm{FAT}}(G)\geq \chi(G)$

but strict inequalities in either direction occur.

3. Spectral Criteria

Let $L=I-D^{-1}A$ be the normalized Laplacian of $G$ with spectrum $0=\lambda_1\leq\cdots\leq\lambda_N\leq 2$, and let $\mu$ be the maximal multiplicity of any eigenvalue.

• Main Theorem: If $G$ admits a FAT $k$-coloring with fairness-parameter $\alpha$, then $\lambda = k\alpha$ is an eigenvalue of $L$ with multiplicity at least $\max\{1, k-1\}$ [(Beers et al., 21 Oct 2025), Thm. 3.4].
• Spectral Bound: This gives

$\chi^{\mathrm{FAT}}(G)\leq \mu+1$

with equality for $K_N$ where $\mu=N-1$ [(Beers et al., 21 Oct 2025), Cor. 3.5].

• Regular graphs: For $d$-regular $G$, $dk\alpha$ is an eigenvalue of the Kirchhoff Laplacian, and $d(1-k\alpha)$ of the adjacency matrix, with multiplicity at least $k-1$ [(Beers et al., 21 Oct 2025), Thm. 3.7].

4. Exact Values for Standard Graph Classes

Some families admit explicit formulas for $\chi^{\mathrm{FAT}}$:

Graph Class	$\chi^{\mathrm{FAT}}(G)$ Formula	Comments
Complete $K_N$	$N$	Each vertex its own class
Bipartite, Tree	$2$	Proper 2-coloring achieves $\alpha=1$, $\beta=0$
Cycle $C_N$	$1$ (if $N$ odd, $3\nmid N$); $2$ (if $N$ even, $3\nmid N$); $3$ (if $3 \mid N$)	See [(Beers et al., 21 Oct 2025), Ex. 2.18]
Petal graph	$3$	$\delta=2$; proper 3-coloring
Book graph $B_m$	$2$ if $m$ odd; $3$ if $m$ even	Odd/even degree gcd governs the value
Turán $T(N,t)$	$\max\{t,N/t\}$	For $t\mid N$, $d$-regular case

Further, in the edgeless graph $\bar{K}_n$, $\chi=1$ and $\chi^{\mathrm{FAT}}=n$ (singleton color classes) (Shaebani, 18 Nov 2025).

5. Relationship with Ordinary Chromatic Number

The FAT chromatic number is not bounded above or below by any function of the ordinary chromatic number $\chi(G)$. Explicit constructions demonstrate the unbounded separation in both directions [(Shaebani, 18 Nov 2025), Thms. 2.1–2.2]:

• There exist families with $\chi(G)$ fixed and $\chi^{\mathrm{FAT}}(G)\to\infty$, and vice versa.
• For example, in the graph $G_1$ formed from $K_{n,n}$ minus a perfect matching, $\chi(G_1)=2$ but $\chi^{\mathrm{FAT}}(G_1)=n$ [(Shaebani, 18 Nov 2025), Thm. 2.1].
• Conversely, for a graph $G_2$ with a large clique and attached pendant triangles, $\chi^{\mathrm{FAT}}(G_2)=2$ while $\chi(G_2)$ is arbitrarily large [(Shaebani, 18 Nov 2025), Thm. 2.2].
• No functions $f, g : \mathbb{N}\to\mathbb{R}$ exist such that $\chi^{\mathrm{FAT}}(G)\leq f(\chi(G))$ or $\chi(G)\leq g(\chi^{\mathrm{FAT}}(G))$ for all $G$.

For disconnected graphs, unbounded gaps are easily achieved by assembling suitable disjoint unions of cliques.

6. Computational Aspects and Algorithmic Considerations

The computational complexity of determining $\chi^{\mathrm{FAT}}(G)$ remains unresolved. Whether this decision problem is NP-complete, or whether it admits efficient polynomial-time algorithms for general or restricted classes, is posed as an open problem [(Beers et al., 21 Oct 2025), Question 7.2]. A constructive result is that, given a FAT $k$-coloring, one can merge classes to obtain FAT $\ell$-colorings $\ell \mid k$ with associated parameters, yielding a full lattice of colorings from the maximal irreducible ones [(Beers et al., 21 Oct 2025), Thm. 5.1].

7. Open Problems and Directions

Significant questions remain regarding the FAT chromatic number, many of which are explicitly stated in the literature [(Beers et al., 21 Oct 2025), §7], (Shaebani, 18 Nov 2025):

1. Gap Quantification: For which classes is $|\chi^{\mathrm{FAT}}(G)-\chi(G)|$ bounded or characterized?
2. Algorithmic Complexity: Is computing $\chi^{\mathrm{FAT}}(G)$ NP-complete?
3. Probabilistic Behavior: What are typical values of $\chi^{\mathrm{FAT}}$ for random regular graphs?
4. Relaxed FAT-variants: What can be said for edge/FAT-variants with only tolerance but not fairness?
5. Monotonicity: Does subgraph monotonicity hold, i.e., for $H \subseteq G$, is $\chi^{\mathrm{FAT}}(H) \leq \chi^{\mathrm{FAT}}(G)$?
6. Spectral Extensions: Can spectral characterization for non-regular graphs be established?
7. Turán Extensions: Can the explicit formula for $\chi^{\mathrm{FAT}}$ on Turán graphs be extended to all multipartite graphs?
8. Enumerative Questions: Which $G$ admit exactly $b$ irreducible FAT colorings for a given $b$?
9. Parameter Realizability: For fixed $k$ and $\alpha\leq 1/(k-1)$, does there exist $G$ with such a FAT $k$-coloring?
10. Simultaneous Maximization: For $k\geq 1$, does there exist $G$ with $\chi^{\mathrm{FAT}}(G) = k$ and parameter $\alpha$ as above?

A plausible implication is that future results on the interplay between $\chi^{\mathrm{FAT}}$ and spectral graph theory may yield deeper insight into equitable partitioning in complex networks.

8. Illustrative Examples

The following table summarizes computed values for small canonical graphs (Beers et al., 21 Oct 2025, Shaebani, 18 Nov 2025):

Graph	$\chi^{\mathrm{FAT}}$	Distinguishing Property
$K_4$	4	Complete graph: each vertex distinct class
$P_n$	2	Path: proper 2-coloring
$C_5$	1	Cycle, $N=5$ (odd, $3\nmid 5$)
$C_6$	3	Cycle, $3\mid 6$
$C_4$	2	Cycle, $N=4$ (even, $3\nmid 4$)
$K_{1,7}$	2	Star (bipartite)

These examples demonstrate the diversity and sometimes counterintuitive values that the FAT chromatic number can take, further underscoring its distinction from the conventional graph chromatic number.

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For further details and proofs, refer to "Fair and Tolerant (FAT) Graph Colorings" by Beers and Mulas (Beers et al., 21 Oct 2025) and "On Fair and Tolerant Colorings of Graphs" by Shaebani (Shaebani, 18 Nov 2025).