Cereceda's conjecture¶
The conjecture that sufficiently many-color proper colorings of a d-degenerate graph can be reconfigured into one another by quadratically many single-vertex recolorings.
Core Idea¶
For a graph of degeneracy d and at least d plus two colors, the coloring-reconfiguration graph should have diameter bounded quadratically in the number of vertices, under the exact threshold convention. Vertices of a meta-graph are proper colorings, edges change one vertex while preserving properness and degeneracy supplies low-degree vertices used to construct bounded recoloring paths. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Cereceda's conjecture belongs to reconfiguration problems and is useful where the analyst can specify the typed reconfiguration problems carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the finite graph and vertex count, degeneracy, color palette size, proper colorings, single-vertex move rule, reconfiguration graph, diameter bound and current proved cases or open status are explicit. The scope is broad within that domain but bounded by the need for the finite graph and vertex count, degeneracy, color palette size, proper colorings, single-vertex move rule, reconfiguration graph, diameter bound and current proved cases or open status are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the finite graph and vertex count, degeneracy, color palette size, proper colorings, single-vertex move rule, reconfiguration graph, diameter bound and current proved cases or open status are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Cereceda's conjecture. Cereceda's conjecture compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed reconfiguration problems carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the finite graph and vertex count, degeneracy, color palette size, proper colorings, single-vertex move rule, reconfiguration graph, diameter bound and current proved cases or open status are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of reconfiguration problems because they reuse the typed reconfiguration problems carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, Vertices of a meta-graph are proper colorings, edges change one vertex while preserving properness and degeneracy supplies low-degree vertices used to construct bounded recoloring paths., and type the carrier, state every parameter and convention in the definition, test that the finite graph and vertex count, degeneracy, color palette size, proper colorings, single-vertex move rule, reconfiguration graph, diameter bound and current proved cases or open status are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Cereceda's conjecture Domain-specific
Parents (1) — more general patterns this builds on
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Cereceda's conjecture is a kind of State and State Transition Prime
The proposed strict upward parent is
prime:state_and_state_transition.
Hierarchy path (1) — routes to 1 parentless root
- Cereceda's conjecture → State and State Transition → Phase Space
Neighborhood in Abstraction Space¶
Cereceda's conjecture sits in a crowded region of the domain-specific corpus (40th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Extremal & Geometric Combinatorics (13 abstractions)
Nearest neighbors
- Reconstruction conjecture — 0.90
- Split graph — 0.89
- Monochromatic triangle — 0.89
- Dissociation number — 0.89
- Transitive reduction — 0.89
Computed from structural-signature embeddings · 2026-09-08