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

TL;DR

This work investigates CAI-partitions (connected acyclic + independent) in Eulerian oriented planar graphs, connecting them to Barnette's Hamiltonicity and Neumann-Lara's dicoloring conjectures. It proves two strong positive results: every planar bipartite 2-vertex-connected subcubic oriented graph and every 2-vertex-connected simple series-parallel graph admit CAI-partitions, using discharging and ear-decomposition techniques, respectively. It also constructs a counterexample showing that relying on the tripartition of Eulerian oriented planar triangulations cannot, in general, guarantee a CAI-partition for the induced subgraphs, highlighting the limitations of this strategy. Together, these results delineate the reach and boundaries of CAI-partitions in oriented planar graphs and suggest directions for extending the approach to broader graph classes or alternative decompositions.

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.

Figures (15)

• Figure 1: \ref{['lem:22']}.
• Figure 2: \ref{['lem:2vC4']}.
• Figure 3: \ref{['lem:232']}.
• Figure 4: \ref{['lem:3C4']}.
• Figure 5: \ref{['lem:2C4']}.

Theorems & Definitions (52)

• Conjecture 1: Barnette's Hamiltonicity Conjecture, 1969 B68
• Conjecture 2: Dual Barnette
• Conjecture 3: Neumann-Lara Dicoloring Conjecture, 1985 NL85
• Conjecture 4: Eulerian Neumann-Lara
• Definition 5: CAI-partition
• Theorem 9
• Theorem 10
• Theorem 11
• Corollary 12
• proof : Proof of \ref{['obs:main']}