Kewen Zhao
Articles written in Proceedings – Mathematical Sciences
Volume 122 Issue 3 August 2012 pp 319-328
Vertex Pancyclicity and New Sufficient Conditions
For a graph $G,\delta(G)$ denotes the minimum degree of 𝐺. In 1971, Bondy proved that, if 𝐺 is a 2-connected graph of order 𝑛 and $d(x)+d(y)\geq n$ for each pair of non-adjacent vertices $x,y$ in 𝐺, then 𝐺 is pancyclic or $G=K_{n/2,n/2}$. In 2001, $Xu$ proved that, if 𝐺 is a 2-connected graph of order $n\geq 6$ and $|N(x)\cup N(y)|+\delta(G)\geq n$ for each pair of non-adjacent vertices $x,y$ in 𝐺, then 𝐺 is pancyclic or $G=K_{n/2,n/2}$. In this paper, we introduce a new sufficient condition of generalizing degree sum and neighborhood union and prove that, if 𝐺 is a 2-connected graph of order $n\geq 6$ and $|N(x)\cup N(y)|+d(w)\geq n$ for any three vertices $x,y,w$ of $d(x,y)=2$ and $wx$ or $wy \notin E(G)$ in 𝐺, then 𝐺 is 4-vertex pancyclic or 𝐺 belongs to two classes of well-structured exceptional graphs. This result also generalizes the above results.
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