A NOTE ON CONNECTIVITY PRESERVING SPLITTING OPERATION FOR MATROIDS REPRESENTABLE OVER $GF(p)$

The splitting operation on a $p$-matroid does not necessarily preserve connectivity. It is observed that there exists a single element extension of the splitting matroid which is connected. In this paper, we define the element splitting operation on $p$-matroids which consist of a splitting operation followed by a single element extension. It is proved that the element splitting operation on a connected $p$-matroid yields a connected $p$-matroid. We give a sufficient condition to yield Eulerian $p$-matroid from Eulerian $p$-matroid under the element splitting operation. A sufficient condition to obtain Hamiltonian $p$-matroid by applying the element splitting operation on $p$-matroid is also provided. The characterization of the paving $p$-matroid which are closed under the element splitting operation, is also obtained.