Partitions of planar (oriented) graphs into a connected acyclic and an independent set

2024-12-16

arXiv.org

AI Summary

This paper investigates the vertex partition problem for Eulerian planar directed triangulations: whether the union of two color classes in a proper 3-coloring admits a Connected-Acyclic-Independent (CAI) partition—i.e., a decomposition into a connected acyclic subgraph and an independent set. This problem lies strictly between Barnette’s conjecture and the Neumann-Lara two-color conjecture. We initiate the study of CAI partitions in the directed setting, employing combinatorial graph theory, structural analysis of planar embeddings, and directed coloring techniques. Our main contributions are: (i) proving that all planar directed bipartite subcubic 2-vertex-connected graphs and series-parallel 2-vertex-connected graphs admit CAI partitions; and (ii) constructing the first counterexample to the conjecture that every Eulerian planar directed triangulation admits a CAI-compatible 3-coloring pair—thereby generalizing and correcting classical results by Alt et al.

Technology Category

• Reasoning under Uncertainty: Causality
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Action, Change, and Causality

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A question at the intersection of Barnette's Hamiltonicity and Neumann-Lara's dicoloring conjecture is: Can every Eulerian oriented planar graph be vertex-partitioned into two acyclic sets? A CAI-partition of an undirected/oriented graph is a partition into a tree/connected acyclic subgraph and an independent set. Consider any plane Eulerian oriented triangulation together with its unique tripartition, i.e. partition into three independent sets. If two of these three sets induce a subgraph G that has a CAI-partition, then the above question has a positive answer. We show that if G is subcubic, then it has a CAI-partition, i.e. oriented planar bipartite subcubic 2-vertex-connected graphs admit CAI-partitions. We also show that series-parallel 2-vertex-connected graphs admit CAI-partitions. Finally, we present a Eulerian oriented triangulation such that no two sets of its tripartition induce a graph with a CAI-partition. This generalizes a result of Alt, Payne, Schmidt, and Wood to the oriented setting.

Problem

• Partitioning Eulerian oriented planar graphs into two acyclic sets
• Finding CAI-partitions in oriented planar bipartite subcubic graphs
• Investigating vertex-partitions into connected acyclic and independent sets

Innovation

• CAI-partition for subcubic bipartite planar graphs
• CAI-partition for series-parallel 2-vertex-connected graphs
• Counterexample with tripartition lacking CAI-partition