Decomposing Split Graphs into Locally Irregular Graphs

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 of G into locally irregular subgraphs. A graph is said to be decomposable if it admits a locally irregular decomposition. In this paper we prove that any decomposable split graph whose clique has at least 10 vertices can be decomposed into at most three locally irregular subgraphs. Furthermore, we characterize those whose decomposition can be into one or two locally irregular subgraphs.