Well-colored graph¶
A graph for which greedy vertex coloring uses the chromatic number of colors under every vertex ordering.
Core Idea¶
The property quantifies over all vertex orders, not one successful greedy run, applies to the declared simple undirected graph convention and is equivalently equality of chromatic and Grundy numbers. Greedy coloring assigns each vertex the least color absent from its already colored neighbors; the graph is well-colored when no ordering can force more colors than an optimal coloring needs. 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¶
Well-colored graph belongs to graph coloring and is useful where the analyst can specify the typed graph coloring carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the finite undirected graph and vertex set, proper vertex coloring, chosen vertex order, greedy first-fit rule, color count for each order, chromatic number, Grundy number, equality of minimum and maximum greedy counts, forbidden or structural examples and hereditary-status qualifications are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the finite undirected graph and vertex set, proper vertex coloring, chosen vertex order, greedy first-fit rule, color count for each order, chromatic number, Grundy number, equality of minimum and maximum greedy counts, forbidden or structural examples and hereditary-status qualifications 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 Well-colored graph. Well-colored graph 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 graph coloring carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the finite undirected graph and vertex set, proper vertex coloring, chosen vertex order, greedy first-fit rule, color count for each order, chromatic number, Grundy number, equality of minimum and maximum greedy counts, forbidden or structural examples and hereditary-status qualifications are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of graph coloring because they reuse the typed graph coloring carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Greedy coloring assigns each vertex the least color absent from its already colored neighbors; the graph is well-colored when no ordering can force more colors than an optimal coloring needs., and type the carrier, state every parameter and convention in the definition, test that the finite undirected graph and vertex set, proper vertex coloring, chosen vertex order, greedy first-fit rule, color count for each order, chromatic number, Grundy number, equality of minimum and maximum greedy counts, forbidden or structural examples and hereditary-status qualifications are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Well-colored graph Domain-specific
Parents (1) — more general patterns this builds on
-
Well-colored graph is a kind of Invariance Prime
The proposed strict upward parent is
prime:invariance.
Hierarchy path (1) — routes to 1 parentless root
- Well-colored graph → Invariance
Neighborhood in Abstraction Space¶
Well-colored graph sits in a crowded region of the domain-specific corpus (1st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Graph Coloring & Labeling (14 abstractions)
Nearest neighbors
- Greedy coloring — 0.97
- Exact coloring — 0.95
- Conflict-free coloring — 0.95
- Equitable coloring — 0.95
- Rainbow coloring — 0.94
Computed from structural-signature embeddings · 2026-09-08