Displaying similar documents to “On the existence of a cycle of length at least 7 in a (1,≤ 2)-twin-free graph”

On long cycles through four prescribed vertices of a polyhedral graph

Jochen Harant, Stanislav Jendrol', Hansjoachim Walther (2008)

Discussiones Mathematicae Graph Theory

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For a 3-connected planar graph G with circumference c ≥ 44 it is proved that G has a cycle of length at least (1/36)c+(20/3) through any four vertices of G.

A Triple of Heavy Subgraphs Ensuring Pancyclicity of 2-Connected Graphs

Wojciech Wide (2017)

Discussiones Mathematicae Graph Theory

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A graph G on n vertices is said to be pancyclic if it contains cycles of all lengths k for k ∈ {3, . . . , n}. A vertex v ∈ V (G) is called super-heavy if the number of its neighbours in G is at least (n+1)/2. For a given graph H we say that G is H-f1-heavy if for every induced subgraph K of G isomorphic to H and every two vertices u, v ∈ V (K), dK(u, v) = 2 implies that at least one of them is super-heavy. For a family of graphs H we say that G is H-f1-heavy, if G is H-f1-heavy for...

Forbidden Pairs and (k,m)-Pancyclicity

Charles Brian Crane (2017)

Discussiones Mathematicae Graph Theory

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A graph G on n vertices is said to be (k, m)-pancyclic if every set of k vertices in G is contained in a cycle of length r for each r ∈ {m, m+1, . . . , n}. This property, which generalizes the notion of a vertex pancyclic graph, was defined by Faudree, Gould, Jacobson, and Lesniak in 2004. The notion of (k, m)-pancyclicity provides one way to measure the prevalence of cycles in a graph. We consider pairs of subgraphs that, when forbidden, guarantee hamiltonicity for 2-connected graphs...

The bondage number of graphs: good and bad vertices

Vladimir Samodivkin (2008)

Discussiones Mathematicae Graph Theory

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The domination number γ(G) of a graph G is the minimum number of vertices in a set D such that every vertex of the graph is either in D or is adjacent to a member of D. Any dominating set D of a graph G with |D| = γ(G) is called a γ-set of G. A vertex x of a graph G is called: (i) γ-good if x belongs to some γ-set and (ii) γ-bad if x belongs to no γ-set. The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph...