Area Requirement and Symmetry Display of Planar Upward Drawings.
R. Tamassia, G. di Battista, I.G. Toms (1992)
Discrete & computational geometry
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R. Tamassia, G. di Battista, I.G. Toms (1992)
Discrete & computational geometry
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Mirko Horňák, Stanislav Jendrol (1996)
Discussiones Mathematicae Graph Theory
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The type of a face f of a planar map is a sequence of degrees of vertices of f as they are encountered when traversing the boundary of f. A set 𝒯 of face types is found such that in any normal planar map there is a face with type from 𝒯. The set 𝒯 has four infinite series of types as, in a certain sense, the minimum possible number. An analogous result is applied to obtain new upper bounds for the cyclic chromatic number of 3-connected planar maps.
Robert E. Tarjan, P. Rosenstiehl (1986)
Discrete & computational geometry
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D.G. Kirkpatrick (1988)
Discrete & computational geometry
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Bjelica, Momčilo (1999)
Novi Sad Journal of Mathematics
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Didimo, Walter (2006)
Journal of Graph Algorithms and Applications
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Brandes, Ulrik, Handke, Dagmar (1998)
Discrete Mathematics and Theoretical Computer Science. DMTCS [electronic only]
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L.J. Schulman (1993)
Discrete & computational geometry
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Dávid Hudák, Peter Šugerek (2012)
Discussiones Mathematicae Graph Theory
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A graph is called 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree δ ≥ 4 contains an edge with degrees of its endvertices of type (4, ≤ 13) or (5, ≤ 9) or (6, ≤ 8) or (7,7). We also show that for δ ≥ 5 these bounds are best possible and that the list of edges is minimal (in the sense that, for each of the considered edge types there are 1-planar graphs whose set of types of edges contains...
Vladimir Batagelj (1989)
Banach Center Publications
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Christina M. Mynhardt, Christopher M. van Bommel (2016)
Discussiones Mathematicae Graph Theory
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A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational weights such that for each edge e of G, the sum of the weights of the triangles that contain e equals 1. We present a necessary and sufficient condition for a planar multigraph to be triangle decomposable. We also show that if a simple planar graph is rationally triangle decomposable,...
Halina Bielak (1999)
Discussiones Mathematicae Graph Theory
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We give a lower bound for the Ramsey number and the planar Ramsey number for C₄ and complete graphs. We prove that the Ramsey number for C₄ and K₇ is 21 or 22. Moreover we prove that the planar Ramsey number for C₄ and K₆ is equal to 17.
D.T. Lee, A.K. Lin (1986)
Discrete & computational geometry
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Kaufmann, Michael, Wiese, Roland (2002)
Journal of Graph Algorithms and Applications
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Izolda Gorgol (2005)
Discussiones Mathematicae Graph Theory
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The planar Ramsey number PR(G,H) is defined as the smallest integer n for which any 2-colouring of edges of Kₙ with red and blue, where red edges induce a planar graph, leads to either a red copy of G, or a blue H. In this note we study the weak induced version of the planar Ramsey number in the case when the second graph is complete.