Every signed planar graph without cycles of length from 4 to 7 is 3-colorable

TL;DR

This paper proves that all signed planar graphs without cycles of length 4 to 7 are 3-colorable, providing a simplified proof and an improved result in the context of signed graph coloring.

Contribution

The authors offer a concise proof that signed planar graphs without cycles of length 4 to 8 are 3-colorable and refine the bounds for cycle lengths.

Findings

Signed planar graphs without cycles of length 4 to 7 are 3-colorable.

Provided a shorter, simpler proof of this coloring property.

Improved the cycle length bound from 8 to 7.

Abstract

Hu and Li investigate the signed graph version of Erd$\ddot{\mathrm{o}}$s problem: Is there a constant $c$ such that every signed planar graph without $k$-cycles, where $4\leq k\leq c$, is $3$-colorable and prove that each signed planar graph without cycles of length from 4 to 8 is 3-colorable. We give a very short and simple proof of this result and improve it, based on a recent observation.