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Projective Plane (projective + plane)
Selected AbstractsOn possible counterexamples to Negami's planar cover conjectureJOURNAL OF GRAPH THEORY, Issue 3 2004Petr Hlin Abstract A simple graph H is a cover of a graph G if there exists a mapping , from H onto G such that , maps the neighbors of every vertex , in H bijectively to the neighbors of , (,) 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, and the first author that the conjecture holds as long as the graph K1,2,2,2 has no finite planar cover. However, those results seem to say little about counterexamples if the conjecture was not true. We show that there are, up to obvious constructions, at most 16 possible counterexamples to Negami's conjecture. Moreover, we exhibit a finite list of sets of graphs such that the set of excluded minors for the property of having finite planar cover is one of the sets in our list. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 183,206, 2004 [source] An algebraic characterization of projective-planar graphsJOURNAL OF GRAPH THEORY, Issue 4 2003Lowell Abrams Abstract We give a detailed algebraic characterization of when a graph G can be imbedded in the projective plane. The characterization is in terms of the existence of a dual graph G* on the same edge set as G, which satisfies algebraic conditions inspired by homology groups and intersection products in homology groups. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 320,331, 2003 [source] Another two graphs with no planar coversJOURNAL OF GRAPH THEORY, Issue 4 2001Petr Hlin Abstract A graph H is a cover of a graph G if there exists a mapping , from V(H) onto V(G) such that , maps the neighbors of every vertex , in H bijectively to the neighbors of ,(,) 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 conjecture holds as long as K1,2,2,2 has no finite planar cover. However, this is still an open question, and K1,2,2,2 is not the only minor-minimal graph in doubt. Let ,,4 (,2) denote the graph obtained from K1,2,2,2 by replacing two vertex-disjoint triangles (four edge-disjoint triangles) not incident with the vertex of degree 6 with cubic vertices. We prove that the graphs ,,4 and ,2 have no planar covers. This fact is used in [P. Hlin,nư, R. Thomas, On possible counterexamples to Negami's planar cover conjecture, 1999 (submitted)] to show that there are, up to obvious constructions, at most 16 possible counterexamples to Negami's conjecture. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 227,242, 2001 [source] Area-minimizing projective planes in 3-manifoldsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2010H. Bray Let (M,g) be a compact Riemannian manifold of dimension 3, and let , denote the collection of all embedded surfaces homeomorphic to \input amssym ${\Bbb R}{ \Bbb P}^2$. We study the infimum of the areas of all surfaces in ,. This quantity is related to the systole of (M,g). It makes sense whenever , is nonempty. In this paper, we give an upper bound for this quantity in terms of the minimum of the scalar curvature of (M,g). Moreover, we show that equality holds if and only if (M,g) is isometric to \input amssym ${\Bbb R}{ \Bbb P}^3$ up to scaling. The proof uses the formula for the second variation of area and Hamilton's Ricci flow. © 2010 Wiley Periodicals, Inc. [source] | ||||||||||