Few Long Lists for Edge Choosability of Planar Cubic Graphs

TL;DR

This paper proves a strengthened edge choosability result for planar cubic graphs, showing that a small set of edges can be assigned 4 colours to ensure proper edge colouring under certain list constraints.

Contribution

It introduces a new bound involving cut-edges for the edge choosability of planar cubic graphs, extending known results.

Findings

Every loopless cubic graph is 4-edge choosable.

For planar cubic graphs with b cut-edges, a set of at most 5b/2 edges can be assigned 4 colours.

Proper edge colouring exists under specified list constraints.

Abstract

It is known that every loopless cubic graph is 4-edge choosable. We prove the following strengthened result. Let G be a planar cubic graph having b cut-edges. There exists a set F of at most 5b/2 edges of G with the following property. For any function L which assigns to each edge of F a set of 4 colours and which assigns to each edge in E(G)-F a set of 3 colours, the graph G has a proper edge colouring where the colour of each edge e belongs to L(e).