On Possible Counterexamples to Negami's Planar Cover Conjecture
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On possible counterexamples to Negami's planar cover conjecture
A simple graph H is a cover of a graph G if there exists a mapping φ from H onto G such that φ maps the neighbors of every vertex v in H bijectively to the neighbors of φ(v) in G. Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. The conjecture is still open. It follows from the results of Archdeacon, Fellows, Negami, a...
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In 1988, Seiya Negami published a conjecture stating that a graph G has a finite planar cover (i.e. a homomorphism from some planar graph onto G which maps the vertex neighbourhoods bijectively) if and only if G embeds in the projective plane. Though the ”if” direction is easy, and over ten related research papers have been published during the past 20 years of investigation, this beautiful con...
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A graph H is a cover of a graph G if there exists a mapping φ from V (H) onto V (G) such that φ maps the neighbors of every vertex v in H bijectively to the neighbors of φ(v) in G. Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. It follows from the results of Archdeacon, Fellows, Negami, and the author that the conjec...
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A graph H is a cover of a graph G if there exists a mapping φ from V (H) onto V (G) such that for every vertex v of G, φ maps the neighbours of v in H bijectively onto the neighbours of φ(v) in G. Negami conjectured in 1987 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. This conjecture is not completely solved yet, but partial results due to A...
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