Displaying similar documents to “Some crossing numbers of products of cycles”

The crossing numbers of products of a 5-vertex graph with paths and cycles

Marián Klešč (1999)

Discussiones Mathematicae Graph Theory

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There are several known exact results on the crossing numbers of Cartesian products of paths, cycles or stars with "small" graphs. Let H be the 5-vertex graph defined from K₅ by removing three edges incident with a common vertex. In this paper, we extend the earlier results to the Cartesian products of H × Pₙ and H × Cₙ, showing that in the general case the corresponding crossing numbers are 3n-1, and 3n for even n or 3n+1 if n is odd.

On the crossing numbers of G □ Cₙ for graphs G on six vertices

Emília Draženská, Marián Klešč (2011)

Discussiones Mathematicae Graph Theory

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The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. The crossing numbers of G☐Cₙ for some graphs G on five and six vertices and the cycle Cₙ are also given. In this paper, we extend these results by determining crossing numbers of Cartesian products G☐Cₙ for some connected graphs G of order six with six and seven edges. In addition, we collect known results concerning crossing numbers of G☐Cₙ for graphs G on six vertices. ...

Heavy Subgraph Conditions for Longest Cycles to Be Heavy in Graphs

Binlong Lia, Shenggui Zhang (2016)

Discussiones Mathematicae Graph Theory

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Let G be a graph on n vertices. A vertex of G with degree at least n/2 is called a heavy vertex, and a cycle of G which contains all the heavy vertices of G is called a heavy cycle. In this note, we characterize graphs which contain no heavy cycles. For a given graph H, we say that G is H-heavy if every induced subgraph of G isomorphic to H contains two nonadjacent vertices with degree sum at least n. We find all the connected graphs S such that a 2-connected graph G being S-heavy implies...

Cycle Double Covers of Infinite Planar Graphs

Mohammad Javaheri (2016)

Discussiones Mathematicae Graph Theory

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In this paper, we study the existence of cycle double covers for infinite planar graphs. We show that every infinite locally finite bridgeless k-indivisible graph with a 2-basis admits a cycle double cover.

Edge cycle extendable graphs

Terry A. McKee (2012)

Discussiones Mathematicae Graph Theory

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A graph is edge cycle extendable if every cycle C that is formed from edges and one chord of a larger cycle C⁺ is also formed from edges and one chord of a cycle C' of length one greater than C with V(C') ⊆ V(C⁺). Edge cycle extendable graphs are characterized by every block being either chordal (every nontriangular cycle has a chord) or chordless (no nontriangular cycle has a chord); equivalently, every chord of a cycle of length five or more has a noncrossing chord.

Disjoint 5-cycles in a graph

Hong Wang (2012)

Discussiones Mathematicae Graph Theory

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We prove that if G is a graph of order 5k and the minimum degree of G is at least 3k then G contains k disjoint cycles of length 5.

Cycles in graphs and related problems

Antoni Marczyk

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Our aim is to survey results in graph theory centered around four themes: hamiltonian graphs, pancyclic graphs, cycles through vertices and the cycle structure in a graph. We focus on problems related to the closure result of Bondy and Chvátal, which is a common generalization of two fundamental theorems due to Dirac and Ore. We also describe a number of proof techniques in this domain. Aside from the closure operation we give some applications of Ramsey theory in the research of cycle...

Strongly pancyclic and dual-pancyclic graphs

Terry A. McKee (2009)

Discussiones Mathematicae Graph Theory

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Say that a cycle C almost contains a cycle C¯ if every edge except one of C¯ is an edge of C. Call a graph G strongly pancyclic if every nontriangular cycle C almost contains another cycle C¯ and every nonspanning cycle C is almost contained in another cycle C⁺. This is equivalent to requiring, in addition, that the sizes of C¯ and C⁺ differ by one from the size of C. Strongly pancyclic graphs are pancyclic and chordal, and their cycles enjoy certain interpolation and extrapolation properties...