Dynamic coloring of graphs having no $K_5$ minor

TL;DR

This paper proves that all simple connected graphs without a $K_5$ minor can be properly 4-colored with a specific neighborhood coloring property, extending known results from planar graphs to a broader class.

Contribution

It generalizes the 4-coloring theorem for planar graphs to all graphs with no $K_5$ minor, introducing a new coloring condition related to neighborhoods.

Findings

Graphs with no $K_5$ minor are 4-colorable with neighborhood constraints.

The result extends the 4-color theorem beyond planar graphs.

Cycle of length 5 is the unique exception.

Abstract

We prove that every simple connected graph with no $K_5$ minor admits a proper 4-coloring such that the neighborhood of each vertex $v$ having more than one neighbor is not monochromatic, unless the graph is isomorphic to the cycle of length 5. This generalizes the result by S.-J. Kim, S. J. Lee, and W.-J. Park on planar graphs.