## A sufficient condition for pancyclic graphs.(Chinese. English summary)Zbl 1110.05055

A graph $$G$$ of order $$n$$ is pancyclic if it contains a cycle of length $$k$$ for each $$k=3,4,\dots, n$$. This paper gives a sufficient condition for a graph to be pancyclic, that is, shows that for any integer $$t\geq 2$$ and a $$2$$-connected graph with order $$n$$ and minimum degree $$\delta\geq t$$, if $$| N(u)\cup N(v)| \geq n-t$$ for any two vertices with distance two in $$G$$, then $$G$$ is pancyclic unless either $$G\cong C_5$$ or $$G\cong K_{n/2,n/2}$$. This improves the result of J. Xu in [Acta Math. Appl. Sin. 24, No. 2, 310–313 (2001; Zbl 1003.05069)].

### MSC:

 05C38 Paths and cycles

### Keywords:

$$2$$-connected graph; minimum degree

Zbl 1003.05069