## Pancyclic graphs and degree sum and neighborhood union involving distance two.(English)Zbl 1241.05059

Discrete Appl. Math. 160, No. 3, 218-223 (2012); retraction ibid. 160, No. 16-17, 2497 (2012).
Summary: For a graph $$G$$, $$\delta$$ denotes the minimum degree of $$G$$. In [J. Comb. Theory, Ser. B 11, 80–84 (1971; Zbl 0183.52301)], J. A. 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 [J. Nanjing Norm. Univ., Nat. Sci. Ed. 29, No. 2, 31–34 (2006; Zbl 1110.05055)], W. Wu, Z. Qi, X. Yuan and Z. Sun proved that, if $$G$$ is a 2-connected graph of order $$n\geq 6$$ and $$|N(x)\cup N(y)|+ \delta\geq n$$ for each pair of non-adjacent vertices $$x$$, $$y$$ of $$d(x,y)= 2$$ in $$G$$, then $$G$$ is pancyclic or $$G= K_{n/2,n/2}$$.
In this paper, we introduce a new condition which generalizes two conditions of 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 $$w x$$ or $$wy\not\in E(G)$$ in $$G$$, then $$G$$ is pancyclic or $$G= K_{n/2,n/2}$$. This result also generalizes the above two results.
Editorial remark: According to the retraction notice, “the original submission was made without the approval of the two previously listed co-authors Ping Zhang and Yu-Jong Tzeng, who neither contributed to the research nor agreed with corresponding author Kewen Zhao to have a paper submitted in their names. This represents a clear violation of our policies on authorship.”

### MSC:

 05C38 Paths and cycles 05C05 Trees 05C12 Distance in graphs

### Keywords:

pancyclic graphs; degree sum; neighborhood union

### Citations:

Zbl 0183.52301; Zbl 1110.05055
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