Published online by Cambridge University Press: 20 November 2018
Let d=(d 1,…,dn ) be a sequence of positive integers. In this note we show that d is realizable by a graph whose square is hamiltonian if and only if (i) d is realizable by some graph, (ii) n≥3, and (iii) d 1+…+d n≥2(n-1). In fact, we prove that if d is realizable by a connected graph, then d is realizable by a graph with a spanning caterpillar. From this it follows that if d is realizable by a connected graph, it is realizable by a graph whose square is pancyclic. We also prove that d is realizable by a graph with a spanning wreath if and only if d is realizable by some graph and d 1+…+dn ≥2n. (A wreath is a connected graph that has exactly one cycle and all vertices not in the cycle monovalent.)