Completely Independent Spanning Trees in Split Graphs: Structural Properties and Complexity

TL;DR

This paper investigates the existence and properties of completely independent spanning trees in split graphs, establishing their relation to hypergraph colorings, and proves the NP-completeness of finding two such trees.

Contribution

It introduces a novel connection between CIST in split graphs and hypergraph colorings, providing bounds and complexity results for their existence.

Findings

Existence of CIST relates to hypergraph colorings.

NP-completeness of finding two CIST in split graphs.

Formulation of a conjecture on hypergraph bipanchromatic number.

Abstract

We study completely independent spanning trees (CIST), \textit{i.e.}, trees that are both edge-disjoint and internally vertex-disjoint, in split graphs. We establish a correspondence between the existence of CIST in a split graph and some types of hypergraph colorings (panchromatic and bipanchromatic colorings) of its associated hypergraph, allowing us to obtain lower and upper bounds on the number of CIST. Using these relations, we prove that the problem of the existence of two CIST in a split graph is NP-complete. Finally, we formulate a conjecture on the bipanchromatic number of a hypergraph related to the results obtained for the number of CIST.