Cycles through specified vertices. (English) Zbl 0780.05033

The authors relax a minimum degree condition on a graph which guarantees long cycles and consider a set \(W\) of vertices with degree at least \(d\geq 1\), in a graph \(G\) with \(n\) vertices in total. Without imposing any further conditions on \(G\), it is shown that there is a cycle in \(G\) containing at least \(\bigl\lceil{| W|\over\lceil n/d\rceil- 1}\bigr\rceil\) vertices in \(W\). Extremal graphs are produced to show that the result is best possible.


05C38 Paths and cycles
05C35 Extremal problems in graph theory
Full Text: DOI


[1] N. Alon: The largest cycle of a graph with a large minimal degree,J. Graph Theory 10 (1986), 123-7. · Zbl 0592.05042
[2] B. Bollobás, andR. Häggkvist: The circumference of a graph with given minimal degree,A Tribute to Paul Erd?s (A. Baker, B. Bollobás and A. Hajnal eds.), Cambridge University Press 1989.
[3] G. A. Dirac: Some theorems on abstract graphs,Proc. London Math. Soc. 2 (1952), 69-81. · Zbl 0047.17001
[4] Y. Egawa, andT. Miyamoto: The longest cycles in a graphG with minimum degree at least|G|/k, J. Combinatorial Theory, Series B 46 (1989), 356-362. · Zbl 0669.05043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.