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arXiv · 1109.4344

Long paths and cycles passing through specified vertices under the average degree condition

Binlong Li·Bo Ning·Shenggui Zhang
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Abstract

Let $G$ be a $k$-connected graph with $k\geq 2$. In this paper we first prove that: For two distinct vertices $x$ and $z$ in $G$, it contains a path passing through its any $k-2$ {specified} vertices with length at least the average degree of the vertices other than $x$ and $z$. Further, with this result, we prove that: If $G$ has $n$ vertices and $m$ edges, then it contains a cycle of length at least $2m/(n-1)$ passing through its any $k-1$ specified vertices. Our results generalize a theorem of Fan on the existence of long paths and a classical theorem of Erd\"os and Gallai on the existence of long cycles under the average degree condition.

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Binlong Li, Bo Ning, Shenggui Zhang. 2011-09-20. Long paths and cycles passing through specified vertices under the average degree condition. https://doi.org/10.1007/s00373-015-1573-y

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