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Chromatic polynomials

1946; American Mathematical Society; Volume: 60; Linguagem: Inglês

10.1090/s0002-9947-1946-0018401-4

1088-6850

George D. Birkhoff, Drew Lewis,

Advanced Mathematical Identities

Four-color reducibility of the four-ring.6. Inequalities satisfied by Ki(\), K1(\), Li(X), £s(X).7. Formula for the reduction of the five-ring in terms of constrained chromatic polynomials.4218. Formula for the reduction of the five-ring in terms of free polynomials.9. Proof of (8.4) using Kempe chains.10.Proof of (8.4) by induction.42511. Four-color reducibility of the five-ring surrounding more than a single region.... 12.Further consequences of the analysis of the five-ring.Chapter VI.Partial analysis of the ra-ring with special attention to the 6-ring and 7-ring 1.The elementary maps and fundamental constrained polynomials entering into the theory of the n-ring.2. The problem of expressing the constrained polynomials in terms of free polynomials.4363. General linear relationships for the fundamental constrained polynomials found by use of Kempe chains.4. Linear inequalities.5. Fundamental linear relations for the six-ring.6.The four-color reducibility of four pentagons surrounding a boundary.7. Further consequences of the partial analysis of the six-ring.8. Fundamental linear relations for the 7-ring.9.The four-color reducibility of three pentagons touching a boundary of a hexagon.Bibliography.4501.The four-ring (Birkhoff [l, p. 120]).three, on the right-hand side.