# Decomposing split graphs into locally irregular graphs

## TL;DR

This paper investigates how split graphs can be broken down into the fewest possible locally irregular subgraphs, providing bounds and characterizations for such decompositions.

## Contribution

It proves that any decomposable split graph can be decomposed into at most three locally irregular subgraphs and characterizes all such graphs based on the number of subgraphs needed.

## Key findings

- Any decomposable split graph can be decomposed into at most three locally irregular subgraphs.
- Complete characterizations for split graphs with 1, 2, or 3 locally irregular subgraph decompositions.
- Identification of conditions under which fewer subgraphs suffice for decomposition.

## Abstract

A graph is locally irregular if any pair of adjacent vertices have distinct degrees. A locally irregular decomposition of a graph $G$ is a decomposition $\mathcal{D}$ of $G$ such that every subgraph $H \in \mathcal{D}$ is locally irregular. A graph is said to be decomposable if it admits a locally irregular decomposition. We prove that any decomposable split graph can be decomposed into at most three locally irregular subgraphs and we characterize all split graphs whose decomposition can be into one, two or three locally irregular subgraphs.