Graphs whose r-neighbourhoods form conformal hypergraphs (Q1061750)

The notion of an open r-neighbourhood of a graph G is examined. The paper aims at describing the structure of those graphs whose open r- neighbourhoods, say \(\pi_ r(G)\), constitute a conformal hypergraph. It is proved that if \(\pi_ r(G)\) is conformal with \(r\geq 1\) then G has no cycle of length \(\leq 3r\) and no two edges of G are incident provided \(r\geq 2\). For a connected graph G, \(\pi_ 1(G)\) is conformal if and only if G has no cycles of lengths 3 and 6.