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The paper introduces a hierarchy of strong chromatic number variants to characterize and algorithmically guarantee SD-EF1, EF1, and EF[1,1] allocations under graph conflict constraints, showing 3Δ-1 agents suffice for any graph of maximum degree Δ.

2026-07-02 04:01 UTC pith:XMWHBK6U

arxiv 2607.01059 v1 pith:XMWHBK6U submitted 2026-07-01 cs.GT cs.DSmath.CO

Fair Allocation under Conflict Constraints via Strong Colorability

Ishay Haviv This is my paper
classification cs.GT cs.DSmath.CO
keywords agentsfairallocationgraphnumbersd-ef1commonconflict
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

In this setting, items are vertices in a graph and conflicts mean adjacent vertices cannot be given to the same agent. All agents share the same preferences over items. The task is to divide the items so the bundles are fair according to three envy-freeness standards that relax full envy-freeness by allowing one extra item. The authors define a ladder of stronger versions of an older graph measure called the strong chromatic number. They prove that the first two rungs of this ladder exactly capture the conditions needed for the strongest fairness notion (SD-EF1) and give sufficient conditions for the other two. A concrete payoff is that any graph whose busiest vertex has at most Δ neighbors can always be divided fairly among 3Δ−1 agents for all three standards. They also supply fast deterministic algorithms that succeed as soon as the number of agents exceeds 3Δ by any positive amount. The approach therefore supplies both existence guarantees and practical computation methods while improving earlier bounds that came from equitable coloring.

Core claim

every graph with maximum degree Δ admits SD-EF1, EF1, and EF[1,1] allocations for common preferences whenever the number of agents is at least 3Δ-1. We further provide, for any ε>0, deterministic polynomial-time algorithms that find such allocations whenever the number of agents is at least (3+ε)Δ

Load-bearing premise

The fairness criteria for agents with common preferences can be reduced to properties of the first two levels of the newly introduced hierarchy of strong chromatic number variants (as described in the abstract).

Editorial analysis

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Circularity Check

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No circularity: results derived from new hierarchy definitions and external graph-theoretic facts

full rationale

The paper introduces a new hierarchy of strong chromatic number variants and reduces the fairness criteria (SD-EF1, EF1, EF[1,1]) to properties of the first two levels of this hierarchy. Existential and algorithmic guarantees for graphs of maximum degree Δ are then obtained from these definitions plus standard graph coloring arguments, extending external results (Alon-Fellows on strong coloring; Barman-Viswanathan on equitable colorings; Equbal et al. on paths). No self-citations are load-bearing, no parameters are fitted and relabeled as predictions, no ansatzes are imported via citation, and no uniqueness theorems from the same authors are invoked. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the newly introduced hierarchy of strong chromatic number variants and standard domain assumptions about graphs and valuations; no data-fitted parameters are present.

axioms (2)
  • domain assumption The input is an undirected simple graph representing conflict constraints
    Standard modeling choice for allocation problems with conflicts, invoked throughout the abstract.
  • domain assumption All agents share identical preferences over the items
    Explicitly stated as the setting for the fairness criteria.
invented entities (1)
  • Hierarchy of variants of the strong chromatic number no independent evidence
    purpose: To provide a unified framework linking graph coloring quantities to the three fairness criteria
    Newly defined in the paper to obtain both existential and algorithmic results.

pith-pipeline@v0.9.1-grok · 5903 in / 1392 out tokens · 86473 ms · 2026-07-02T04:01:45.400080+00:00 · methodology

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In the fair allocation problem under conflict constraints, the goal is to partition the vertices of a graph among agents in a fair manner, such that no two adjacent vertices are assigned to the same agent. We study this problem for agents with common preferences through the lens of three fairness criteria: stochastic-dominance envy-freeness up to one item for preference orders (SD-EF1), envy-freeness up to one item for monotone additive valuations (EF1), and envy-freeness up to one item from each side for general additive valuations (EF[1,1]). To do so, we introduce a hierarchy of variants of the strong chromatic number, a graph quantity introduced independently by Alon and Fellows in the early nineties. Our results reveal a close connection between fair allocation under conflict constraints and the first two levels of this hierarchy, providing a unified route to both existential and algorithmic results. For SD-EF1, we fully characterize the number of agents needed to guarantee a fair allocation of a given graph for every common preference order. For EF1 and EF[1,1], we provide analogous sufficient conditions, extending a result on path graphs due to Equbal, Gurjar, Igarashi, Kumar, Manurangsi, Nath, Saxena, Vaish, and Yoneda. We also show that, unlike in the SD-EF1 setting, the sufficient conditions for EF1 and EF[1,1] are not necessary in general. Our framework yields existential and algorithmic consequences in terms of the maximum degree. We obtain that every graph with maximum degree $\Delta$ admits SD-EF1, EF1, and EF[1,1] allocations for common preferences whenever the number of agents is at least $3\Delta-1$. We further provide, for any $\varepsilon>0$, deterministic polynomial-time algorithms that find such allocations whenever the number of agents is at least $(3+\varepsilon)\Delta$. These guarantees strengthen earlier work by Barman and Viswanathan on equitable colorings.

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