Displaying similar documents to “NP-completeness results for minimum planar spanners.”

The list linear arboricity of planar graphs

Xinhui An, Baoyindureng Wu (2009)

Discussiones Mathematicae Graph Theory

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The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. An and Wu introduce the notion of list linear arboricity lla(G) of a graph G and conjecture that lla(G) = la(G) for any graph G. We confirm that this conjecture is true for any planar graph having Δ ≥ 13, or for any planar graph with Δ ≥ 7 and without i-cycles for some i ∈ {3,4,5}. We also prove that ⌈½Δ(G)⌉ ≤ lla(G) ≤ ⌈½(Δ(G)+1)⌉ for any planar graph having Δ ≥ 9. ...

A new proof of the four-colour theorem.

Robertson, Neil, Sanders, Daniel P., Seymour, Paul, Thomas, Robin (1996)

Electronic Research Announcements of the American Mathematical Society [electronic only]

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Planar Ramsey numbers

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.

On properties of maximal 1-planar graphs

Dávid Hudák, Tomáš Madaras, Yusuke Suzuki (2012)

Discussiones Mathematicae Graph Theory

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A graph is called 1-planar if there exists a drawing in the plane so that each edge contains at most one crossing. We study maximal 1-planar graphs from the point of view of properties of their diagrams, local structure and hamiltonicity.

On An Extremal Problem In The Class Of Bipartite 1-Planar Graphs

Július Czap, Jakub Przybyło, Erika Škrabuľáková (2016)

Discussiones Mathematicae Graph Theory

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A graph G = (V, E) is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite 1-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most 3n − 8 edges, where n denotes the order of a graph. We show that maximal-size bipartite 1-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same...

Light edges in 1-planar graphs with prescribed minimum degree

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...

Characterizations of planar plick graphs

V.R. Kulli, B. Basavanagoud (2004)

Discussiones Mathematicae Graph Theory

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In this paper we present characterizations of graphs whose plick graphs are planar, outerplanar and minimally nonouterplanar.