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

Mohammed Lalou, Nader Mbarek, Abdallah Skender, Olivier Togni

TL;DR

This work analyzes completely independent spanning trees (CIST) in split graphs by establishing a deep link to hypergraph colorings. It shows that the existence of k CIST corresponds to panchromatic colorings of the associated hypergraph, while bipanchromatic colorings yield constructive CIST under certain conditions, enabling bounds on the maximum number of CIST via the hypergraph parameters $\chi_p(H)$, $\chi_p^2(H)$, and $\alpha_k(H)$. The authors prove NP-completeness for deciding the existence of two CIST in split graphs and provide ILP formulations to explore the conjectured relationships between panchromatic and bipanchromatic colorings. These results contribute tightness-oriented bounds and a framework for future algorithmic and complexity investigations in CIST and hypergraph coloring contexts.

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.

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

TL;DR

This work analyzes completely independent spanning trees (CIST) in split graphs by establishing a deep link to hypergraph colorings. It shows that the existence of k CIST corresponds to panchromatic colorings of the associated hypergraph, while bipanchromatic colorings yield constructive CIST under certain conditions, enabling bounds on the maximum number of CIST via the hypergraph parameters , , and . The authors prove NP-completeness for deciding the existence of two CIST in split graphs and provide ILP formulations to explore the conjectured relationships between panchromatic and bipanchromatic colorings. These results contribute tightness-oriented bounds and a framework for future algorithmic and complexity investigations in CIST and hypergraph coloring contexts.

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.

Paper Structure

This paper contains 12 sections, 16 theorems, 5 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. CIST and hypergraph coloring
  4. Properties related to unique colors
  5. Relation between CIST and unique colors
  6. Bounds on the number of CIST in split graphs
  7. Complexity
  8. Concluding remarks
  9. Linear modeling
  10. The panchromatic coloring problem
  11. The bipanchromatic coloring problem
  12. The minimum number of unique colors problem

Key Result

Theorem 1

Let $T_1, \dots, T_k$ be spanning trees in a graph $G$. Then, $T_1, \dots , T_k$ are completely independent if and only if $T_1, \dots , T_k$ are edge-disjoint and for any vertex $x \in V(G)$, there is at most one spanning tree $T_i$ such that $d_{T_i}(x) > 1.$

Figures (12)

  • Figure 1: A split graph and its corresponding hypergraph
  • Figure 2: A split graph without $3$ CIST whose corresponding hypergraph is panchromatically $3$-colorable
  • Figure 3: A split graph having $2$ CIST while not satisfying the conditions of Theorem \ref{['thm3']}
  • Figure 4: A split graph with $2$ CIST whose corresponding hypergraph is not bipanchromatically $2$-colorable
  • Figure 5: A split graph $G$ where $x_D$ is adjacent to all vertices of $I$ but its color is not unique in some of the panchromatic $2$-colorings of $H(G)$
  • ...and 7 more figures

Theorems & Definitions (27)

  • Theorem 1: 2
  • Theorem 2: 5
  • Theorem 3: 29
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 17 more