On automorphisms of line-graphs (Q1149454)

This paper generalizes some results on hypergraph reconstruction due to \textit{C. Berge} [C. R. Acad. Sci., Paris, Ser. A 274, 1783-1786 (1972; Zbl 0236.05129)] and \textit{J.C.Fournier} [Proc. 1rst Working Sem. Hypergraphs, Columbus 1972, Lecture Notes Math. 411, 95-98 (1974; Zbl 0302.05113)]. Let \(\alpha\) be an automorphism of the line-graph of the \(r\)-uniform hypergraph \(H\) with \(n\) points. If the valencies of \(H\) \(v(x_1)\leq v(x_2)\leq\dots\leq v(x_n)\) and \(v(n,r)\) and \(v(x_{2r})>v(n,r)=\binom{n-1}{r-1}-\binom{n-r-1}{r-1}+1\), then for \(n>4r\alpha\) is induced by an automorphism of \(H\) (i.e. a permutation of \(V(H)\)). Two examles show that the valency conditions of above theorem cannot be weakened in any point of \(H\). The proof uses a result of \textit{A.J.W.Hilton} and \textit{E.C.Milner} [Q. J. Math., Oxf. II. Ser. 18, 369-384 (1967; Zbl 0168.26205)] sharpening the \textit{P.Erdős, Chao Ko, R.Rado} [Q. J. Math., Oxf. II. Ser. 12, 313-320 (1961; Zbl 0100.01902)] theorem to find the maximal intersection subfamilies of the hypergraph \(H\).