Abstract In this paper we define some new labellings for trees, called the in-improper and out-improper odd-graceful labellings such that some trees labelled with the new labellings can induce graceful graphs having at least a cycle. We, next, apply the new labellings to construct large scale of graphs having improper graceful/odd-graceful labellings or having graceful/odd-graceful labellings.

Congenital malalignment of the great toenails is a disorder in which malalignment of the nail matrix results in angular lateral nail plate growth. The consequence is onychodystrophy, which may progress to onychogryphosis and chronic ingrown toenails. There has been debate as to whether the condition is inherited or acquired in utero secondary to improper fetal positioning. A 6-year-old boy and ...

This is a complement to the comparison theorem for improper integrals in the textbook. The vanilla version presented in the textbook is good enough to solve some very easy examples and it becomes exponentially gory with the complexity of the integral. Fortunately it is not hard to refine the statement in the book, and turn it into a powerful tool to estimate the convergence of arbitrarily compl...