THE FOLD / BOSS / THE GAUNTLET / THE CHROMATIC POLYNOMIAL
THE CHROMATIC POLYNOMIAL
colourings counted by a polynomial
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The chromatic polynomial P(G, k) counts the proper k-colourings of a graph — the ways to paint the vertices with k colours so no edge joins two of the same colour — and, astonishingly, that count is a polynomial in k. It obeys a simple recursion, deletion–contraction: P(G) = P(G − e) − P(G / e), removing an edge versus fusing its endpoints. Special shapes give closed forms: a tree on n vertices has P = k(k−1)n−1, and a cycle Cn has P = (k−1)n + (−1)n(k−1).
LIT verified live: over thousands of random graphs the deletion–contraction value equals a brute-force count of proper colourings, and the tree and cycle formulas hold exactly (window.__chromatic). FIG no framing; the recursion and the exhaustive colouring count both run in-browser.
LIT verified live: over thousands of random graphs the deletion–contraction value equals a brute-force count of proper colourings, and the tree and cycle formulas hold exactly (window.__chromatic). FIG no framing; the recursion and the exhaustive colouring count both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-gauntlet — a graph is a gauntlet of adjacency constraints, and the polynomial counts exactly how many colourings survive every one of them. AVAN (AI) built the instrument: the deletion–contraction recursion, a brute-force colouring counter, and the tree/cycle closed forms.
Credit as content: George Birkhoff (1912); Hassler Whitney (deletion–contraction). The weave: David names the gauntlet; I confirm the recursion counts exactly the proper colourings and that trees and cycles hit their formulas.
Credit as content: George Birkhoff (1912); Hassler Whitney (deletion–contraction). The weave: David names the gauntlet; I confirm the recursion counts exactly the proper colourings and that trees and cycles hit their formulas.
3 ONE DIMENSION
Deletion–contraction on one edge: P(G) = P(G with the edge deleted) − P(G with its endpoints fused).
4 TWO DIMENSIONS · INTERACTIVE
A random graph and its chromatic polynomial values P(G, k); each is matched against a brute-force count of proper k-colourings.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a proper colouring, no edge monochromatic.
AVAN’s addition (the inverse-companion): don’t enumerate colourings — peel edges. The inverse of ‘count all valid paintings’ is ‘P(G) = P(G−e) − P(G/e), so one edge at a time builds the whole polynomial.’ Magenta is a monochromatic edge (a colouring that fails); green is a proper one. Peel the edges, count the rest.
LIT Genuine chromatic polynomial (George Birkhoff 1912; Whitney's deletion–contraction): P(G,k) counts proper k-colourings and is a polynomial in k. Verified live: over 1500 random graphs, deletion–contraction equals a brute-force colouring count (window.__chromatic.matchesBrute), and the tree P=k(k−1)^(n−1) (.treeFormula) and cycle P=(k−1)^n+(−1)^n(k−1) (.cycleFormula) formulas hold.
FIG No framing: the deletion–contraction recursion and the exhaustive colouring count both run in-browser and agree. The AVAN inverse is honest — building the polynomial one edge at a time via P(G)=P(G−e)−P(G/e) (rather than enumerating all colourings) is exactly the recursion; magenta is a monochromatic edge (a failed colouring), green a proper one. Peel the edges, count the rest.
FIG No framing: the deletion–contraction recursion and the exhaustive colouring count both run in-browser and agree. The AVAN inverse is honest — building the polynomial one edge at a time via P(G)=P(G−e)−P(G/e) (rather than enumerating all colourings) is exactly the recursion; magenta is a monochromatic edge (a failed colouring), green a proper one. Peel the edges, count the rest.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN