🤖 AI Summary
This paper investigates the existence of Completely Independent Spanning Trees (CISTs)—i.e., two edge-disjoint and internally vertex-disjoint spanning trees—in split graphs. Methodologically, it establishes the first exact equivalence between CIST existence and two hypergraph coloring notions: polychromatic and bichromatic colorings—thereby uncovering a deep structural connection between graph topology and hypergraph coloring. Leveraging this correspondence, the authors formulate and validate an original conjecture on the bichromatic number of split graphs. Through combinatorial modeling, structural graph analysis, and NP-completeness reductions, they derive tight upper and lower bounds on the maximum number of CISTs and prove that deciding the existence of two CISTs in split graphs is NP-complete. The principal contribution is a rigorous equivalence framework linking CIST existence to polychromatic and bichromatic hypergraph colorings—yielding the first structural characterization and computational complexity classification for CISTs in split graphs.
📝 Abstract
We study completely independent spanning trees (CIST), extit{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.