Summary: This paper is concerned with oscillation of the second-order halflinear delay dynamic equation
on a time scale , where is the quotient of odd positive integers, , and are positive rd-continuous functions on , is positive and (delta) differentiable, , and . We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results in the special cases when and involve and improve some oscillation results for second-order differential and difference equations; and when , and our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.