On Barnette's Conjecture and $H^{+-}$ property

TL;DR

This paper investigates conditions under which certain plane triangulations can be partitioned into two trees, providing partial solutions related to Barnette's conjecture and the $H^{+-}$ property.

Contribution

It proves that if specific induced subgraphs are acyclic, then particular vertex partitions into trees are possible in simple even plane triangulations.

Findings

Partitioning into two trees is possible under given acyclicity conditions.

Results relate to Barnette's conjecture and bipartite cubic plane graphs.

Provides conditions for vertex partitions containing specific paths.

Abstract

A conjecture of Barnette states that every 3-connected cubic bipartite plane graph has a Hamilton cycle, which is equivalent to the statement that every simple even plane triangulation admits a partition of its vertex set into two subsets so that each induces a tree. Let $G$ be a simple even plane triangulation and suppose that ${V_1, V_2, V_3}$ is a 3-coloring of the vertex set of $G$. Let $B_{i}$, $i = 1, 2, 3$, be the set of all vertices in $V_i$ of the degree at least 6. We prove that if induced graphs $G[B_1 \cup B_2]$ and $G[B_1 \cup B_3]$ are acyclic, then the following properties are satisfied: [6pt] (1) For every path $abc$ there is possible to partition the vertex set of $G$ into two subsets so that each induces a tree, and one of them contains the edge $ab$ and avoids the vertex $c$, [6pt] (2) For every path $abc$ with vertices $a$, $c$ of the same color there is possible to partition the vertex set of $G$ into two subsets so that each induces a tree, and one of them contains the path $abc$.