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arxiv: 2604.21665 · v1 · submitted 2026-04-23 · 🧮 math.CO · math.SP

Spectral Perspectives on FAT Graph Colorings

Lies Beers , Raffaella Mulas This is my paper

Pith reviewed 2026-05-09 21:13 UTC · model grok-4.3

classification 🧮 math.CO math.SP
keywords FAT chromatic numbercomplete multipartite graphsspectral methodsgraph coloringgraph operationscombinatorial argumentsadjacency matrixfair and tolerant colorings
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The pith

The FAT chromatic number is exactly determined for every complete multipartite graph.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper explores Fair and Tolerant (FAT) graph colorings, a framework in which each vertex shares its color with a prescribed fraction of neighbors while distributing the remaining neighbors evenly among the other color classes. The central result determines the FAT chromatic number for all complete multipartite graphs. Spectral methods supply the primary proofs, supported by combinatorial arguments, and the work tracks how FAT colorings behave under standard graph operations.

Core claim

We determine the FAT chromatic number for all complete multipartite graphs. Spectral methods form the primary focus, with several combinatorial arguments included to complement the results. We also analyze the behavior of FAT colorings under several graph operations.

What carries the argument

The FAT coloring condition, requiring each vertex to share its color with a fixed fraction of neighbors and balance the rest evenly across other classes, whose minimum color count is computed via spectral bounds from the adjacency matrix.

If this is right

  • The minimum number of colors needed for any complete multipartite graph under the FAT rules is now known explicitly.
  • FAT colorings of disjoint unions and joins can be built directly from the colorings of the component graphs.
  • Spectral techniques supply matching upper and lower bounds that coincide exactly for complete multipartite graphs.
  • Combinatorial constructions verify the spectral results in special cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The exact values supply a benchmark for testing whether spectral methods work on FAT colorings of graph families outside the multipartite case.
  • The analysis of operations suggests that the FAT property remains controllable when graphs are built from multipartite pieces.
  • The link between eigenvalue bounds and the even-distribution requirement may carry over to other equitable or balanced coloring problems.

Load-bearing premise

The FAT coloring definition applies uniformly and spectral methods produce the exact chromatic number for these graphs without further unstated restrictions.

What would settle it

A specific complete multipartite graph for which the number of colors given by the spectral formula fails to produce an assignment meeting the share-fraction and even-distribution rules for every vertex.

Figures

Figures reproduced from arXiv: 2604.21665 by Lies Beers, Raffaella Mulas.

Figure 1
Figure 1. Figure 1: An illustration of the concepts of fairness and tolerance. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Tur´an graph T(13, 4). Definition 3.2. The Tur´an graph T(N, t) (see [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A monochromatic (left) and balanced (right) FAT coloring of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A FAT coloring with χ F AT = 2 colors of the complete multipartite graph K6,4,4,2. Proof. We distinguish different cases according to the value of p. If p = 1, then G is a regular Tur´an graph, and the assertion follows directly from Theorem 3.4. Now assume that p > 1. Note that the eigenvalues N/(N − ni) for i = 1, . . . , p cannot give rise to a FAT coloring, since the associated eigenspace does not have… view at source ↗
Figure 5
Figure 5. Figure 5: A FAT 2-coloring of the graph K6,6,1,1,1,1,1,1,1,1,1 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The Tur´an graph T(6, 3) before and after the removal of a coloring class corresponding to two distinct FAT colorings. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Three FAT colorings of C4 ⊔ C4 ⊔ C4 and the corresponding induced coloring of its complement. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Lifting a 2-coloring of C6 to its Cartesian (top), tensor (middle), and strong (bottom) products with K2. It follows from the previous identities, together with Remark 7.3, that all four matrices A(G1□G2), A(G1 ⊠G2), D−1A(G1□G2), and D−1A(G1 ⊠G2) admit a common eigenbasis consisting of ten￾sor products f1 ⊗ f2, where fi is an eigenfunction of A(Gi), D−1A(Gi), and L(Gi) for i = 1, 2. Consequently, the norma… view at source ↗
Figure 9
Figure 9. Figure 9: Lifting a 3-coloring of C6 to its Cartesian (top), tensor (middle), and strong (bottom) products with K2. Theorem 7.5. Let V1, . . . , Vk denote the coloring classes of G1 with respect to a FAT k-coloring with parameter α. Then, the tensor product G1×G2 also admits a FAT k-coloring with parameter α. Moreover, the corresponding coloring classes in G1 × G2 are the sets Vi × V (G2). Proof. By Theorem 2.9, the… view at source ↗
read the original abstract

We investigate Fair and Tolerant (FAT) graph colorings, a coloring framework in which each vertex is allowed to share its color with a prescribed fraction of its neighbors, while the remaining neighbors are required to be distributed evenly among the other coloring classes. In particular, we determine the FAT chromatic number for all complete multipartite graphs, and we analyze the behavior of FAT colorings under several graph operations. Although spectral methods form the primary focus, several combinatorial arguments are included to complement the results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper introduces Fair and Tolerant (FAT) graph colorings, allowing each vertex to share its color with a prescribed fraction of neighbors while requiring even distribution of the remaining neighbors among other color classes. It determines the exact FAT chromatic number for all complete multipartite graphs by applying the known spectrum of their adjacency matrices, supported by explicit combinatorial constructions that achieve the bound for arbitrary part sizes and tolerance values. The work also analyzes the behavior of FAT colorings under several standard graph operations.

Significance. If the derivations hold, the exact formula for the FAT chromatic number on complete multipartite graphs is a useful contribution to relaxed coloring theory. The primary reliance on spectral methods is well-suited to the regular structure of these graphs, and the explicit combinatorial constructions provide constructive verification of the spectral bounds. This combination of spectral derivations and matching constructions is a strength that could support extensions to other graph families or tolerance parameters.

minor comments (2)
  1. The introduction would benefit from a brief illustrative example of a FAT coloring on a small complete multipartite graph (e.g., K_{2,2,2}) to clarify the interaction between the fraction parameter and the even-distribution requirement.
  2. Notation for the tolerance parameters (fraction of same-color neighbors and even distribution) is introduced clearly but could be cross-referenced more explicitly when the main formula is stated in the spectral section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly identifies the core contribution: an exact determination of the FAT chromatic number for all complete multipartite graphs via the known spectrum of their adjacency matrices, together with matching explicit combinatorial constructions. We are pleased that the combination of spectral and constructive arguments is viewed as a strength.

Circularity Check

0 steps flagged

Derivation is self-contained using known spectra and independent constructions

full rationale

The paper explicitly defines FAT colorings with parameters for allowed same-color neighbor fractions and even distribution across classes. It computes the FAT chromatic number for complete multipartite graphs by invoking the standard, externally known spectrum of their adjacency matrices and then verifies the bound via direct combinatorial constructions that work for arbitrary part sizes and tolerance parameters. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology; the spectral input is a standard fact independent of this work, and the constructions are explicit and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are described. Standard graph theory background is implicitly assumed.

axioms (1)
  • standard math Standard axioms of graph theory and linear algebra for spectral methods
    Required for any graph coloring and eigenvalue analysis; invoked implicitly throughout.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

  1. [1]

    Beers and R

    L. Beers and R. Mulas. At the end of the spectrum: Chromatic bounds for the largest eigenvalue of the normalized Laplacian.Journal of Physics: Complexity, 6(2), 2024

    work page 2024

  2. [2]

    Beers and R

    L. Beers and R. Mulas. Fair and Tolerant (FAT) Graph Colorings.arXiv:2510.18494, 2025

    work page arXiv 2025

  3. [3]

    Bosek, J

    B. Bosek, J. Grytczuk, and G. Jak´ obczak. Majority coloring game.Discrete Applied Mathematics, 255:15–20, 2019

    work page 2019

  4. [4]

    Brouwer and W.H

    A.E. Brouwer and W.H. Haemers.Spectra of graphs. Universitext. Springer, New York, 2012

    work page 2012

  5. [5]

    S. Butler. Algebraic aspects of the normalized Laplacian. InRecent Trends in Combina- torics, pages 295–315. Springer, 2016. 24

    work page 2016

  6. [6]

    Chung.Spectral graph theory

    F. Chung.Spectral graph theory. American Mathematical Soc., 1997

    work page 1997

  7. [7]

    J. Jost, R. Mulas, and D. Zhang.Spectra of Discrete Structures.Cambridge University Press, Cambridge Studies in Advanced Mathematics, Forthcoming, 2026

    work page 2026

  8. [8]

    Shaebani

    S. Shaebani. Concerning FAT Colorings of Graphs.arXiv:2512.11153, 2025

    work page arXiv 2025

  9. [9]

    Shaebani

    S. Shaebani. On Fair and Tolerant Colorings of Graphs.arXiv:2511.14871, 2025

    work page arXiv 2025

  10. [10]

    Sun and K.C

    S. Sun and K.C. Das. Normalized Laplacian spectrum of complete multipartite graphs. Discrete Appl. Math., 284:234–245, 2020

    work page 2020

  11. [11]

    B. Zelinka. On domatic numbers of graphs.Mathematica Slovaca, 31(1):91–95, 1981. 25

    work page 1981