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On critically hamiltonian graphs

On critically hamiltonian graphs Acta Mathematica Academiae Scientiarum Hungaricae Tomus 29 (3--4), (1977), pp. 255--258. By LINDA LESNIAK-FOSTER (Baton Rouge) A hamiltonian cycle (hamiltonian path) of a graph G is a cycle (path) containing all the vertices of G. A graph possessing a hamiltonian cycle is called a hamihonian graph. If v is a vertex of a graph G, then G-v denotes the subgraph of G with vertex set V(G)-{v} and whose edges are all those of G not incident with v. A graph G is called critically hamiltonian if G is hamiltonian and for each vertex v of G, the graph G-v is not hamiltonian. Thus, for example, every cycle is a critically hamiltonian graph. It is known (see [1]) that if a graph G of order p=>4 has at least (p~-3p+ 8)/2 edges, then G is hamiltonian and G-v is hamiltonian for each vertex v of G. Therefore the size (number of edges) of a critically hamil- tonian graph of order p_->4 is clearly less than (p~-3p+8)/2. The purpose of this paper is to establish several results concerning the size of critically hamiltonian graphs of given order. In this regard, the following 1emma will be useful. LEMMA. If G is a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungaricae Springer Journals

On critically hamiltonian graphs

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01895844
Publisher site
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Abstract

Acta Mathematica Academiae Scientiarum Hungaricae Tomus 29 (3--4), (1977), pp. 255--258. By LINDA LESNIAK-FOSTER (Baton Rouge) A hamiltonian cycle (hamiltonian path) of a graph G is a cycle (path) containing all the vertices of G. A graph possessing a hamiltonian cycle is called a hamihonian graph. If v is a vertex of a graph G, then G-v denotes the subgraph of G with vertex set V(G)-{v} and whose edges are all those of G not incident with v. A graph G is called critically hamiltonian if G is hamiltonian and for each vertex v of G, the graph G-v is not hamiltonian. Thus, for example, every cycle is a critically hamiltonian graph. It is known (see [1]) that if a graph G of order p=>4 has at least (p~-3p+ 8)/2 edges, then G is hamiltonian and G-v is hamiltonian for each vertex v of G. Therefore the size (number of edges) of a critically hamil- tonian graph of order p_->4 is clearly less than (p~-3p+8)/2. The purpose of this paper is to establish several results concerning the size of critically hamiltonian graphs of given order. In this regard, the following 1emma will be useful. LEMMA. If G is a

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Acta Mathematica Academiae Scientiarum Hungaricae – Springer Journals

Published: Aug 23, 2013

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