# Plane graphs without 4- and 5-cycles and without ext-triangular 7-cycles are 3-colorable

## TL;DR

This paper proves that certain classes of planar graphs, specifically those without 4- and 5-cycles and with restrictions on 7- and 8-cycles, are 3-colorable, advancing understanding related to Steinberg's conjecture.

## Contribution

It establishes 3-colorability for plane graphs without 4- and 5-cycles under the absence of ext-triangular 7-cycles, extending known results.

## Key findings

- Plane graphs without 4- and 5-cycles are 3-colorable if they lack ext-triangular 7-cycles.
- Planar graphs without 4-, 5-, 7-cycles are 3-colorable.
- Planar graphs without 4-, 5-, 8-cycles are 3-colorable.

## Abstract

Listed as No. 53 among the one hundred famous unsolved problems in [J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, Berlin, 2008] is Steinberg's conjecture, which states that every planar graph without 4- and 5-cycles is 3-colorable. In this paper, we show that plane graphs without 4- and 5-cycles are 3-colorable if they have no ext-triangular 7-cycles. This implies that (1) planar graphs without 4-, 5-, 7-cycles are 3-colorable, and (2) planar graphs without 4-, 5-, 8-cycles are 3-colorable, which cover a number of known results in the literature motivated by Steinberg's conjecture.