Discussiones Mathematicae
Graph Theory 18(1) (1998) 5-13
DOI: https://doi.org/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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Received 23 September 1994
Revised 27 November 1996
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