Every planar graph without adjacent short cycles is 3-colorable

TL;DR

This paper proves that planar graphs with no adjacent short cycles, where the sum of lengths of any two adjacent cycles is at least 11, are always 3-colorable, extending understanding of graph coloring constraints.

Contribution

It establishes a new sufficient condition for 3-colorability in planar graphs based on cycle adjacency and length sum constraints.

Findings

Planar graphs with no adjacent cycles summing to less than 11 in length are 3-colorable.

The result generalizes previous coloring theorems by considering cycle adjacency.

Provides a new criterion for graph coloring based on cycle length and adjacency.

Abstract

Two cycles are {\em adjacent} if they have an edge in common. Suppose that $G$ is a planar graph, for any two adjacent cycles $C_{1}$ and $C_{2}$, we have $|C_{1}| + |C_{2}| \geq 11$, in particular, when $|C_{1}| = 5$, $|C_{2}| \geq 7$. We show that the graph $G$ is 3-colorable.