A note on connectivity preserving splitting operation for matroids representable over $GF(p)$

TL;DR

This paper introduces an element splitting operation for $p$-matroids that preserves connectivity and provides conditions for resulting matroids to be Eulerian or Hamiltonian, advancing understanding of matroid connectivity and structure.

Contribution

It defines a new element splitting operation on $p$-matroids that guarantees connectivity preservation and offers conditions for Eulerian and Hamiltonian properties.

Findings

Element splitting on connected $p$-matroids preserves connectivity.

Sufficient conditions for Eulerian $p$-matroids after splitting.

Sufficient conditions for Hamiltonian $p$-matroids after splitting.

Abstract

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 is a splitting operation followed by a single element extension. It is proved that element splitting operation on connected $p$-matroid yields a connected $p$-matroid. We give a sufficient condition to yield Eulerian $p$-matroids from Eulerian $p$-matroids under the element splitting operation. A sufficient condition to obtain hamiltonian $p$-matroid by applying element splitting operation on $p$-matroid is also provided.