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

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.

💡 Research Summary

This research paper provides a rigorous investigation into the structural properties and computational complexity of Completely Independent Spanning Trees (CIST) within the context of split graphs. A CIST is defined as a set of spanning trees that are both edge-disjoint and internally vertex-disjoint, representing a highly constrained form of graph decomposition. The core objective of the study is to establish a mathematical bridge between the existence of these trees in split graphs and specific coloring properties of associated hypergraphs.

The authors begin by defining a mapping from a split graph $G = (D \cup I, E)$—where $D$ is a clique and $I$ is an independent set—to a hypergraph $H(G) = (D, \mathcal{E})$. In this construction, each vertex in the independent set $I$ is treated as a hyperedge in $H(G)$, consisting of its neighbors in $D$. This transformation allows the complex problem of tree construction to be reframed as a hypergraph coloring problem.

A significant portion of the paper is dedicated to distinguishing between two types of hypergraph colorings: panchromatic and bipanchromatic. The authors prove in Theorem 4 that the existence of $k$ CISTs in a split graph implies the existence of a panchromatic $k$-coloring in the associated hypergraph, where every hyperedge contains at least one vertex of each of the $k$ colors. However, they demonstrate that panchromatic coloring alone is insufficient to guarantee CIST existence, providing a counterexample to show that the condition is merely necessary, not sufficient.

To bridge this gap, the authors introduce the concept of “bipanchromatic $k$-coloring.” This refined coloring requires that each color appears at least twice within every hyperedge. Theorem 5 establishes that a bipanchromatic $k$-coloring is a sufficient condition for the existence of $k$ CISTs. The proof involves a constructive method where trees are first built within the clique $D$ using the redundant color instances, and then expanded into spanning trees by appropriately distributing the vertices from the independent set $I$.

The paper also delves into the impact of “unique colors”—colors that appear only once in a hyperedge. Through Lemmas 1 and 2, the authors show that the presence of such vertices creates structural bottlenecks. Specifically, Proposition 1 proves that if a vertex in $D$ is associated with a unique color in a hyperedge, it becomes impossible to form $k$ CISTs because that vertex cannot serve as an internal vertex for multiple trees without violating the vertex-disjoint constraint. This leads to a generalized impossibility criterion based on the minimum number of unique colors, $\alpha_k$.

Finally, the paper addresses the computational complexity of the problem. By performing a polynomial-time reduction from the SET SPLITTING problem to the panchromatic coloring problem, and subsequently to the CIST existence problem in split graphs, the authors prove that determining whether two CISTs exist is NP-complete. This result highlights the inherent difficulty of the problem and provides a definitive complexity classification. In conclusion, the paper significantly advances the field by providing new theoretical tools, such as bipanchromatic coloring, to estimate the bounds of CIST existence and establishing a profound connection between graph theory and hypergraph coloring.