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Distance-Locally Disconnected Graphs

100%

Miller M., Ryan J., Ryjáček Z.

Discussiones Mathematicae Graph Theory

2013

tom 33

nr 1

203-215

For an integer k ≥ 1, we say that a (finite simple undirected) graph G is k-distance-locally disconnected, or simply k-locally disconnected if, for any x ∈ V (G), the set of vertices at distance at least 1 and at most k from x induces in G a disconnected graph. In this paper we study the asymptotic behavior of the number of edges of a k-locally disconnected graph on n vertices. For general graphs, we show that this number is Θ(n2) for any fixed value of k and, in the special case of regular graphs, we show that this asymptotic rate of growth cannot be achieved. For regular graphs, we give a general upper bound and we show its asymptotic sharpness for some values of k. We also discuss some connections with cages.

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Some applications of pq-groups in graph theory

75%

Exoo G.

Discussiones Mathematicae Graph Theory

2004

tom 24

nr 1

109-114

We describe some new applications of nonabelian pq-groups to construction problems in Graph Theory. The constructions include the smallest known trivalent graph of girth 17, the smallest known regular graphs of girth five for several degrees, along with four edge colorings of complete graphs that improve lower bounds on classical Ramsey numbers.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

1 Exoo G.

1 Miller M.

1 Ryan J.

1 Ryjáček Z.

1 2013

1 2004

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Distance-Locally Disconnected Graphs

Some applications of pq-groups in graph theory