The Geng-Hua Fan conditions for pancyclic or Hamilton-connected graphs
Let n,k\(\geq 3\) and \(m\geq 2\) be integers. Denote by F(n,k,m) the family of graphs such that \(G\in F(n,k,m)\) if and only if G is m-connected graph of order n satisfying that for all vertices x, y of G, \(d(x,y)=2\Rightarrow Max(d(x),d(y))\geq k/2,\) where d(x,y) is the distance between x and y in G. \textit{G. Fan} has shown that if \(G\in F(n,k,2)\) then the circumference of G is at least Min(n,k) [J. Comb. Theory, Ser. B 37, 221-227 (1984; Zbl 0551.05048)]. In this paper, the following three results are obtained: (1) If \(G\in F(n,n-1,2)\) and \(\alpha\) (G)\(\leq n/2\), then G is Hamiltonian with some exceptions, where \(\alpha\) (G) is the independence number of G; (2) If \(G\in F(n,n,2)\) then G is pancyclic with some exceptions; (3) If \(G\in F(n,n+1,3)\) then G is Hamilton-connected. Note. The results (2) and (3) above are also obtained by the reviewer and \textit{R. Shi} in J. Syst. Sci. Math. Sci. 6, 258-262 (1986; Zbl 0608.05050).
scientific article; zbMATH DE number 3991544