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

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Wide W.

Discussiones Mathematicae Graph Theory

2017

tom 37

nr 2

477-499

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 every graph H ∈H. Let D denote the deer, a graph consisting of a triangle with two disjoint paths P3 adjoined to two of its vertices. In this paper we prove that every 2-connected {K1,3, P7, D}-f1-heavy graph on n ≥ 14 vertices is pancyclic. This result extends the previous work by Faudree, Ryjáček and Schiermeyer.

A Fan-Type Heavy Pair Of Subgraphs For Pancyclicity Of 2-Connected Graphs

100%

Wideł W.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 1

173-184

Let G be a graph on n vertices and let H be a given graph. We say that G is pancyclic, if it contains cycles of all lengths from 3 up to n, and that it 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 [...] min⁡{dG(u),dG(v)}≥n+12 $\min \{ d_G (u),d_G (v)\} \ge {{n + 1} \over 2}$ . In this paper we prove that every 2-connected {K1,3, P5}-f1-heavy graph is pancyclic. This result completes the answer to the problem of finding f1-heavy pairs of subgraphs implying pancyclicity of 2-connected graphs.

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2 Discussiones Mathematicae Graph Theory

1 Wide W.

1 Wideł W.

1 2017

1 2016

Wyniki wyszukiwania

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

A Fan-Type Heavy Pair Of Subgraphs For Pancyclicity Of 2-Connected Graphs