Cycles in squares of trees without generalized claws

Let the generalized claw S r , r , r arise by subdividing each edge of the claw K 1 , 3 exactly r − 1 times. Harary and Schwenk proved that the square of a tree of order at least 3 has a Hamiltonian cycle if and only if it does not contain S 2 , 2 , 2 . In the present paper we study the circumference of the square of a tree that does not contain S r , r , r for some fixed r ≥ 3 . For r = 3 , we give exact bounds and characterize all extremal trees.