Map Color Theorem

Ringel, Gerhard

doi:10.1007/978-3-642-65759-7

Cited by 373 publications

(231 citation statements)

References 22 publications

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“…This landmark result conjectured by Heawood [11] was proved by Ringel [18] and Ringel-Youngs [17]. Furthermore, every graph with chromatic number H(g) embedded on Σ contains a complete graph on H(g) vertices as a subgraph.…”

Section: Graphs On Surfacesmentioning

confidence: 80%

Nordhaus–Gaddum‐type Theorems for decompositions into many parts

Füredi

Kostochka

Škrekovski

et al. 2005

Journal of Graph Theory

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Section: Graphs On Surfacesmentioning

confidence: 80%

Nordhaus–Gaddum‐type Theorems for decompositions into many parts

Füredi

Kostochka

Škrekovski

et al. 2005

Journal of Graph Theory

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“…Graphs on an orientable or non-orientable surface are also called maps. Maps were studied mainly in the context of map coloring theorems [19], [44].…”

Section: Ribbon Graphs and Möbius Graphsmentioning

confidence: 99%

Non-commutative matrix integrals and representation varieties of surface groups in a finite group

Mulase¹,

Yu²

2005

Annales De L’institut Fourier

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A graphical expansion formula for non-commutative matrix integrals with values in a finite-dimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their non-orientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the topological type of the surface on which the graph is drawn. As an example, we calculate the integral on the group algebra of a finite group. We show that the integral is a generating function of the number of homomorphisms from the fundamental group of an arbitrary closed surface into the finite group. The graphical expansion formula yields a new proof of the classical theorems of Frobenius, Schur and Mednykh on these numbers.

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“…Hence a graph embeds in the projective plane if and only if it contains no subdivision of 103 graphs in [6]. Also, a complete graph K n is projective if n = 5 or 6, and the only projective complete bipartite graphs are K 3,3 and K 3,4 (see [4] or [12]). Note that a planar graph is not considered as a projective graph.…”

Section: Introductionmentioning

confidence: 99%