Theoretical Computer Science
cc.complexity-theory graph-theory np-hardness graph-colouring
Updated Mon, 20 Jun 2022 10:21:38 GMT

# Complexity of edge coloring in planar graphs

3-edge coloring of cubic graphs is $NP$-complete. Four Color Theorem is equivalent to "Every cubic planar bridgeless graphs is 3-edge colorable".

What is the complexity of 3-edge coloring of cubic planar graphs?

Also, It is conjectured that $\Delta$-edge coloring is $NP$-hard for planar graphs with maximum degree $\Delta \in${4,5}.

Has any progress been made towards resolving this conjecture?

Marek Chrobak and Takao Nishizeki. Improved edge-coloring algorithms for planar graphs. Journal of Algorithms, 11:102-116, 1990

## Solution

Every bridgeless planar cubic graph can be 3-edge-colored in quadratic time, as this task is equivalent to four-coloring a planar graph, which can be done in quadratic time. (See Robertson, Sanders, Seymour and Thomas: http://people.math.gatech.edu/~thomas/OLDFTP/fcdir/fcstoc.ps )

EDIT: As Mathieu points out, cubic graphs with bridges are never 3-edge colourable.

• +1 – To be precise, the equivalence between $3$-edge coloration and $4$-coloration stands only for bridgeless cubic planar graphs. — May 05, 2011 at 21:18