On the characterization of Eulerian $es$-splitting $p$-matroids

TL;DR

This paper introduces and characterizes the $es$-splitting operation for $p$-matroids, identifying conditions under which it produces Eulerian matroids and analyzing its effects on connectivity and Hamiltonicity.

Contribution

It extends the $es$-splitting operation to $p$-matroids, providing a characterization of when it yields Eulerian matroids and how it affects connectivity and Hamiltonian properties.

Findings

$es$-splitting can produce Eulerian $p$-matroids under certain conditions.

Connectivity and 3-connectedness are preserved by $es$-splitting.

Conditions for Hamiltonian $p$-matroids after $es$-splitting are established.

Abstract

The $es$-splitting operation on binary bridge-less matroids never produces an Eulerian matroid. But for matroids representable over $GF(p),(p>2),$ called $p$-matroids, the $es$-splitting operation may yield Eulerian matroids. In this work, we introduce the $es$-splitting operation for $p$-matroids and characterize a class of $p$-matroids yielding Eulerian matroids after the $es$-splitting operation. Characterization of circuits, and bases of the resulting matroid, after the $es$-splitting operation, in terms of circuits, and bases of the original matroid, respectively, are discussed. We also proved that the $es$-splitting operation on $p$-matroids preserves connectivity and 3-connectedness. Sufficient condition to obtain Hamiltonian $p$-matroid from Hamiltonian $p$-matroid under $es$-splitting operation is also provided.