Abstract
For a graph G, we denote by p(G) and c(G) the number of vertices of a longest path and a longest cycle in G, respectively. For a vertex v in G, id(v) denotes the implicit degree of v. In this paper, we obtain that if G is a 2-connected graph on n vertices such that the implicit degree sum of any three independent vertices is at least n + 1, then either G contains a hamiltonian path, or c(G) ≥ p(G) − 1.
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The first author is supported by the National Natural Science Foundation of China (Grant No. 11501322), the Postdoctoral Science Foundation of China (Grant No. 2015M571999) and the Natural Science Foundation of Shandong Province (Grant No. ZR2014AP002).
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Cai, Jq., Li, H. An implicit degree condition for relative length of long paths and cycles in graphs. Acta Math. Appl. Sin. Engl. Ser. 32, 365–372 (2016). https://doi.org/10.1007/s10255-016-0561-1
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DOI: https://doi.org/10.1007/s10255-016-0561-1