For a graph $G$ let $σ_3(G)$ denote the minimum sum of the degrees of 3 independent vertices and $ρ^*_3 (G)$ the minimum cardinality of the union of the neighborhoods of 3 independent vertices that have at least one common adjacency. It is shown that any 1-tough graph of order $n$ with $σ_3(G)\ge n$ will have a circumference at least the minimum of $n$ and $2ρ^*_3(G)+ 4$. This result extends several Hamiltonian results involving degree and neighborhood conditions and toughness.
R.Faudree (Memphis)