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arxiv: 2606.31702 · v1 · pith:6SXI43RJnew · submitted 2026-06-30 · 🧮 math.CO

List-Coloring and Chromatic-Choosability -- A Dynamic Survey

Nandana K Vasudevan , K Somasundaram , N Narayanan This is my paper

Pith reviewed 2026-07-01 04:30 UTC · model grok-4.3

classification 🧮 math.CO
keywords list-coloringchromatic-choosabilitygraph coloringchoosabilityembedded graphsperfect graphsclaw-free graphsline graphs
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The pith

This survey reviews where list chromatic number equals chromatic number and where gaps remain across major graph classes.

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

This paper surveys developments in list-coloring and chromatic-choosability since the 2001 Woodall survey. It examines graph classes where the list chromatic number equals the chromatic number and classes where a nontrivial gap remains. The review highlights principal proof methods and covers topics including embedded graphs, perfect graphs, complete bipartite and multipartite graphs, claw-free graphs, line graphs, powers of graphs, graph products, and variants of list-coloring. A sympathetic reader would care because these results clarify when graphs remain colorable even when each vertex is restricted to its own list of colors.

Core claim

The survey updates knowledge on list-coloring, where each vertex receives its own list of admissible colors, and chromatic-choosability as the property that this list chromatic number equals the ordinary chromatic number. It organizes results by graph classes with known equality, classes exhibiting nontrivial gaps, and the main methods used to establish them, spanning embedded graphs, perfect graphs, complete bipartite and multipartite graphs, claw-free graphs, line graphs, powers of graphs, graph products, and selected variants.

What carries the argument

Chromatic-choosability, the equality between a graph's list chromatic number and its chromatic number, used to classify families of graphs.

If this is right

  • Equality between list chromatic number and chromatic number holds for some embedded graphs.
  • A nontrivial gap between the two numbers persists for certain complete bipartite and multipartite graphs.
  • Chromatic-choosability is known for perfect graphs and for claw-free graphs under the reviewed conditions.
  • Principal methods developed for line graphs and graph powers extend to related coloring problems.
  • Selected variants of list-coloring build directly on the equality and gap classifications.

Where Pith is reading between the lines

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

  • The survey's organization of proof methods could guide attempts to close gaps in unclassified graph families.
  • Results on graph products might suggest patterns for choosability in larger composite structures.
  • The covered techniques for embedded graphs could be tested on surfaces or embeddings not yet examined in detail.

Load-bearing premise

That the authors have selected and summarized the key results and methods from the literature in a representative and accurate manner without significant omissions or interpretive errors relative to the body of work since the 2001 survey.

What would settle it

A major published result on chromatic-choosability or list-coloring for one of the covered graph classes that is omitted from the survey or presented with a clear factual error.

read the original abstract

List-coloring, introduced independently by Vizing and by Erd\H{o}s, Rubin, and Taylor in the 1970s, generalizes ordinary vertex coloring by assigning to each vertex its own set of admissible colors. A graph is chromatic-choosable if its list chromatic number equals its chromatic number. The previous survey on list-coloring by D R Woodall (2001), emphasized defective choosability, the list-coloring conjectures, and different methods used for list-coloring. This survey reviews major developments on list-coloring and chromatic-choosability, with emphasis on graph classes for which equality is known, graph classes exhibiting a nontrivial gap, and the principal methods used to prove such results. The survey covers embedded graphs, perfect graphs, complete bipartite and multipartite graphs, claw-free graphs, line graphs, powers of graphs, graph products, and selected variants of list-coloring.

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 / 1 minor

Summary. This is a survey paper that updates the 2001 survey by D.R. Woodall on list-coloring. It reviews major developments on list-coloring and chromatic-choosability, emphasizing graph classes where the list chromatic number equals the chromatic number, classes with nontrivial gaps between them, and the principal methods used to prove such results. The coverage includes embedded graphs, perfect graphs, complete bipartite and multipartite graphs, claw-free graphs, line graphs, powers of graphs, graph products, and selected variants of list-coloring.

Significance. If the summaries of results and methods are accurate and representative of the literature since 2001, the survey would serve as a useful reference for researchers working on choosability questions, consolidating progress across multiple graph classes in one place. The focus on equality cases versus gaps and on proof techniques aligns with the needs of the field.

minor comments (1)
  1. The abstract refers to this as a 'dynamic survey' but does not specify in the provided text how updates will be handled or whether a version history or companion website is planned; this could be clarified in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the survey and for recommending acceptance. We are pleased that the focus on equality cases, gaps, and proof techniques is viewed as aligning with the needs of the field.

Circularity Check

0 steps flagged

Survey with no derivations, predictions, or self-referential claims

full rationale

This is a literature survey summarizing results on list-coloring and chromatic-choosability from external sources since the 2001 Woodall survey. It advances no new theorems, equations, fitted parameters, or predictions of its own. The central claim is a representative overview of known equalities, gaps, and proof methods across graph classes, with all substantive content drawn from cited prior work. No step reduces by construction to the paper's own inputs, and no uniqueness theorems or ansatzes are imported from the authors' prior publications. The manuscript is therefore self-contained against external benchmarks with no detectable circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a survey paper, the central contribution is the organization of existing results rather than new derivations or postulates. No free parameters, axioms, or invented entities are introduced by the paper itself.

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Reference graph

Works this paper leans on

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