, Volume 13, Issue 2, pp 147-155

Cycles through specified vertices

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Abstract

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 \(\left\lceil {\frac{w}{{\left\lceil {{n \mathord{\left/ {\vphantom {n d}} \right. \kern-0em} d}} \right\rceil - 1}}} \right\rceil \) vertices ofW. We also find the extremal graphs for this property.

Research supported in part by NSF Grant DMS 8806097