Cycles through specified vertices


Recently, various authors have obtained results about the existence of long cycles in graphs with a given minimum degreed. We extend these results to the case where only some of the vertices are known to have degree at leastd, and we want to find a cycle through as many of these vertices as possible. IfG is a graph onn vertices andW is a set ofw vertices of degree at leastd, we prove that there is a cycle through at least\(\left\lceil {\frac{w}{{\left\lceil {{n \mathord{\left/ {\vphantom {n d}} \right. \kern-\nulldelimiterspace} d}} \right\rceil - 1}}} \right\rceil \) vertices ofW. We also find the extremal graphs for this property.

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Research supported in part by NSF Grant DMS 8806097

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Bollobás, B., Brightwell, G. Cycles through specified vertices. Combinatorica 13, 147–155 (1993).

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AMS subject classification code (1991)

  • 05 C 38