Abstract
Let \(G\) be a \(k\)-connected graph with \(k\ge 2\). In this paper we first prove that: For two distinct vertices \(x\) and \(z\) in \(G\), it contains a path connecting \(x\) and \(z\) which passes 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 Erdős and Gallai on the existence of long cycles under the average degree condition.
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This paper was supported by the NSFC (11271300) and the project NEXLIZ-CZ.1.07/2.3.00/30.0038, which is co-financed by the European Social Fund and the state budget of the Czech Republic.
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Li, B., Ning, B. & Zhang, S. Long Paths and Cycles Passing Through Specified Vertices Under the Average Degree Condition. Graphs and Combinatorics 32, 279–295 (2016). https://doi.org/10.1007/s00373-015-1573-y
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DOI: https://doi.org/10.1007/s00373-015-1573-y