Graph coloring

Mathematical statement

Check every edge in the complete graph. Nonadjacent vertices may share a color. One proper coloring supplies only a chromatic-number upper bound; optimality needs a separate lower-bound certificate, and a greedy result is not automatically optimal.

Representative formula (exact TeX)

C(G)

Assumptions

• The finite undirected simple graph and a color value for every vertex are complete.
• Renaming color labels does not change the underlying vertex partition.

Symbol scope

• C — proper vertex coloring
• G — finite simple graph

Intuition, definition, and misconceptions

The endpoints of every edge have different colors.

The number of used colors is at most k.

An upper bound is not necessarily the chromatic optimum.

All vertices need not have different colors, and a greedy color count is not automatically the minimum.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Finite simple undirected graphs — The complete edge set is explicit.
• Function mapping — A coloring is a total map from V to color labels.
• Natural number — k is an upper bound on the number of colors used.

Verification boundary

Only the specified coloring and upper bound are checked; optimality is not claimed without a lower bound.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add chromatic number and clique lower bounds.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.