Fair and Tolerant Vertex Coloring

• Fair and tolerant vertex coloring is defined by assigning colors to vertices so that a controlled fraction of same-colored neighbors is allowed, ensuring equitable color distribution.
• The framework utilizes real parameters (α and β) to balance color assignments, leading to new structural and spectral bounds that extend classical chromatic concepts.
• This approach bridges combinatorial topology and spectral graph theory, opening research directions in algorithmic complexity and invariant separations compared to proper vertex coloring.

A Fair and Tolerant (FAT) vertex coloring is a framework in graph theory that generalizes traditional coloring concepts by permitting a controlled fraction of neighbors of the same color for each vertex, under a fairness constraint on how all colors are seen in neighbor sets. The FAT chromatic number $\chi^{\mathrm{FAT}}(G)$ quantifies the largest number of colors for which a graph $G$ admits a FAT coloring. Unlike proper vertex coloring, where adjacency imposes strict color separation, FAT colorings interpolate between proper and imbalanced color assignments by encoding tolerance levels via real parameters. This notion has yielded new structural and spectral bounds, motivated separation results for classic invariants, and grounded new directions in combinatorial and algebraic graph theory (Shaebani, 18 Nov 2025, Beers et al., 21 Oct 2025).

1. Formal Definition and Principal Properties

Given a simple finite graph $G=(V,E)$ and an arbitrary vertex coloring $c\colon V\to \{1,\dots,k\}$ with resulting nonempty color classes $V_i = c^{-1}(i)$ $(i=1,\dots,k)$, a FAT $k$-coloring is a coloring for which there exist real parameters $\alpha, \beta \in [0,1]$ such that for every vertex $v\in V$ and every color class $V_i$,

$|V_i \cap N(v)| = \begin{cases} \alpha\,\deg(v), & \text{if $v \notin V_i$} \ \beta\,\deg(v), & \text{if $v \in V_i$} \end{cases}$

where $N(v)$ denotes the open neighborhood of $v$. The parameters must fulfill the balance equation $\beta + (k-1)\alpha = 1$ to partition each neighbor set exactly. The FAT chromatic number is then

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

This encoding recovers proper colorings for $\beta=0$ and $\alpha=1/(k-1)$, but admits much more flexible configurations. In regular graphs, FAT colorings enforce balanced color-class sizes (Shaebani, 18 Nov 2025, Beers et al., 21 Oct 2025).

2. Illustrative Examples and Behavioral Extremes

The theory is informed by explicit computations for graph families:

• Complete graphs $K_n$: Each singleton class $V_i=\{v_i\}$ gives $\chi^{\rm FAT}(K_n) = n$ with $(\alpha,\beta) = (0,1)$.
• Empty graphs $\overline{K_n}$: The trivial partition yields $$\chi^{\mathrm{FAT}}(\overline{K_n})=n \gg \chi(\overline{K_n})=1$$, illustrating a maximal gap.
• Bipartite graphs and trees: For $K_{m,n}$ or trees, $\chi^{FAT}(G) \leq \min(\delta)+1$, often achieved with standard bipartition.
• Cycles $C_n$: For even $n$, a FAT-2-coloring exists; for odd $n$, one can realize FAT-3-colorings, but not two, unless $3 \mid n$ (Beers et al., 21 Oct 2025, Shaebani, 18 Nov 2025).

The essential distinction from classic chromatic number emerges in these bounds and achievable gaps, as detailed in the following table:

Graph Class	$\chi(G)$	$\chi^{\mathrm{FAT}}(G)$	Achievable $(\alpha,\beta)$
$K_n$	$n$	$n$	$(0,1)$
$\overline{K_n}$	$1$	$n$	$(0,1)$
$K_{1,7}$	$2$	$2$	$(1/7,6/7)$
$C_6$ (cycle)	$2$	$2$	$(1/2,0)$

No universal function bounds $\chi^{\mathrm{FAT}}$ in terms of $\chi$ or vice versa, as both absolute and relative gaps can be unbounded (Shaebani, 18 Nov 2025).

3. Structural and Spectral Bounds

FAT colorings admit nontrivial structural and spectral bounds:

• Minimum-degree bound: $\chi^{\mathrm{FAT}}(G)\leq \delta+1$, where $\delta$ is the minimum degree. Each color beyond the own class must be represented in the neighborhood, forcing strict degree constraints.
• Spectral (Laplacian) bound: If $G$ admits a FAT $k$-coloring with parameter $\alpha$, then $\lambda = k\alpha$ is an eigenvalue of the normalized Laplacian $L=I-D^{-1}A$ of $G$, with multiplicity at least $k-1$. Therefore, the FAT chromatic number is bounded above by the maximum eigenvalue multiplicity plus one: $\chi^{\mathrm{FAT}}(G) \leq \mu + 1$.
• Regular graphs: In any $d$-regular connected graph, all color classes must be equal-sized; thus, $k$ divides $|V|$.

A consequence is that FAT colorings are tightly connected to partitioning eigenstructures and to divisibility properties in regular graphs (Beers et al., 21 Oct 2025).

4. Separation from Classical Chromatic Theory

FAT colorings break several expected relations with the standard chromatic number:

• There is no function $f$ such that $\chi^{\mathrm{FAT}}(G) \leq f(\chi(G))$ for all $G$. Disconnected empty graphs or bipartite families with growing colorings (by Beers and Mulas) show $\chi(G)$ fixed while $\chi^{\mathrm{FAT}}(G)$ is unbounded.
• Conversely, no function $g$ satisfies $\chi(G) \leq g(\chi^{\mathrm{FAT}}(G))$ universally; disjoint unions of cliques can force $\chi(G)$ arbitrarily larger than $\chi^{\mathrm{FAT}}(G)$.
• Both differences $\chi^{\mathrm{FAT}}(G)-\chi(G)$ and $\chi(G)-\chi^{\mathrm{FAT}}(G)$ are unbounded, even for connected graphs (Shaebani, 18 Nov 2025).

These separations demarcate FAT coloring as a genuinely distinct invariant with non-monotonic relationships to existing parameters.

5. Algorithmic Aspects and Complexity

The primary existence proofs for FAT colorings rely on combinatorial-topological principles and are often non-constructive. Tucker's lemma, as used in the fair splitting of colored paths, forms a combinatorial analog of the Borsuk–Ulam theorem, applying to sign-vectors encoding color assignments and removals (Alishahi et al., 2017).

• General algorithmic status: Determining $\chi^{\mathrm{FAT}}(G)$ remains an open complexity problem and may be computationally hard. The exponential size of the coloring configuration space (via $k$ and assignments) complicates direct search.
• Special cases via flow/matching: For paths and special graphs, once removal sets are fixed (tolerant splitting), standard bipartite matching or network flow reduces the instance to polynomial time (Alishahi et al., 2017).
• Random and structural colorings: Algorithmic behavior for variants (list-FAT, fractional-FAT) and for random graphs is largely unexplored (Beers et al., 21 Oct 2025).

6. Related Notions: Tolerant Splitting

FAT colorings generalize to partitioning tasks on colored paths:

• Tolerant fair splitting: Every vertex-colored path can be split into two independent sets after removing at most one vertex per color, achieving nearly balanced counts for each color class. The proof applies the octahedral Tucker lemma to encode parity constraints and fair splitting conditions into sign-vectors mapped by symmetric labelings (Alishahi et al., 2017).
• Algorithmic reduction: After a deletion set per color is fixed, finding the fair bipartition reduces to matching, making it tractable for certain path-like inputs.

Such tolerant variants demonstrate the flexibility of the FAT coloring paradigm and its compatibility with combinatorial topology and equitable partitioning results.

7. Open Problems and Research Directions

Current research directions in FAT coloring include:

• Determining $\chi^{\mathrm{FAT}}(G)$ for additional graph families (e.g., hypercubes, chordal, planar, random graphs).
• Investigating the algorithmic complexity for the existence of FAT $k$-colorings and related decision problems.
• Extending structural and spectral bounds, especially for non-regular graphs and via fractional or randomized colorings.
• Completing the classification of irreducible FAT colorings and their coarsenings.
• Clarifying under what containment conditions $\chi^{\mathrm{FAT}}(H) \leq \chi^{\mathrm{FAT}}(G)$ when $H \subseteq G$ (Beers et al., 21 Oct 2025, Shaebani, 18 Nov 2025).

The theory of fair and tolerant vertex coloring, positioned at the intersection of equitable graph partitioning, spectral theory, and combinatorial topology, continues to evolve with foundational and algorithmic advancements.