DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory  18(1) (1998)   5-13
DOI: 10.7151/dmgt.1059

LONG CYCLES AND NEIGHBORHOOD UNION IN 1-TOUGH GRAPHS WITH LARGE DEGREE SUMS

Vu Dinh Hoa

Wundtstr. 7/4L1
01217 Dresden, Germany

Abstract

For a 1-tough graph G we define σ3(G) = min{ d(u)+d(v)+ d(w):{u,v,w} is an independent set of vertices} and NCσ3-n+5(G) = max{i = 1σ3-n+5N(vi):{v1, ..., vσ3-n+5 } is an independent set of vertices}. We show that every 1-tough graph with σ3(G) ≥ n contains a cycle of length at least min{ n, 2NCσ3-n+5(G)+2}. This result implies some well-known results of Faßbender [2] and of Flandrin, Jung & Li [6]. The main result of this paper also implies that c(G) ≥ min{n,2NC2(G)+2} where NC2(G) = min{|N(u) ∪N(v)|:d(u,v) = 2}. This strengthens a result that c(G) ≥ min{n, 2NC2(G)} of Bauer, Fan and Veldman [3].

Keywords: graphs, neighborhood, toughness, cycles.

1991 Mathematics Subject Classification: 05C38, 05C45.

References

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[2] B. Faß, A sufficient condition on degree sums of independent triples for hamiltonian cycles in 1-tough graphs, Ars Combinatoria 33 (1992) 300-304.
[3] D. Bauer, A. Morgana, E. Schmeichel and H.J. Veldman, Long cycles in graphs with large degree sums, Discrete Mathematics 79 (1989/90) 59-70, doi: 10.1016/0012-365X(90)90055-M.
[4] D. Bauer, H.J. Broersma and H.J. Veldman, Around three lemmas in hamiltonian graph theory, in: R. Bodendiek and R. Henn, eds., Topics in Combinatorics and Graph Theory. Festschrift in honour of Gerhard Ringel, Physica-Verlag, Heidelberg (1990) 101-110.
[5] D. Bauer, G. Fan and H.J. Veldman, Hamiltonian properties of graphs with large neighborhood unions, Discrete Mathematics 96 (1991) 33-49, doi: 10.1016/0012-365X(91)90468-H.
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[7] H.J. Broersma, J. Van den Heuvel and H.J. Veldman, Long Cycles, Degree sums and Neighborhood Unions, Discrete Mathematics 121 (1993) 25-35, doi: 10.1016/0012-365X(93)90534-Z.
[8] J. Van den Heuvel, Degree and Toughness Condition for Cycles in Graphs, Thesis (1994) University of Twente, Enschede Niederland.

Received 23 September 1994
Revised 27 November 1996