Efficient Recognition of Subgraphs of Planar Cubic Bridgeless Graphs

TL;DR

This paper presents an efficient algorithm to recognize whether a given planar graph can be extended to a planar cubic bridgeless graph, which is key for 3-edge-colorability, using advanced graph decomposition techniques.

Contribution

It introduces a novel $O(n^2)$-time algorithm for augmenting planar graphs to cubic bridgeless supergraphs or determining impossibility, leveraging the Generalized Antifactor problem and SPQR-trees.

Findings

Algorithm successfully recognizes augmentable graphs within quadratic time.

Provides a constructive method for augmentation or proof of impossibility.

Enhances understanding of 3-edge-colorability in planar graphs.

Abstract

It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs, and hence 3-edge-colorable. We provide an efficient recognition algorithm that given an $n$-vertex planar graph, augments this graph in $O(n^2)$ steps to a planar cubic bridgeless supergraph, or decides that no such augmentation is possible. The main tools involve the Generalized Antifactor-problem for the fixed embedding case, and SPQR-trees for the variable embedding case.