A note on 3-colorable plane graphs without 5- and 7-cycles

TL;DR

This paper revises a previous proof regarding 3-colorable plane graphs without 5- and 7-cycles by fixing a gap and clarifying the definitions and explanations, ensuring the proof's correctness.

Contribution

It corrects and clarifies a prior proof on 3-colorability of certain plane graphs by refining definitions and adding explanations.

Findings

The corrected proof confirms 3-colorability under specified conditions.

Clarification of special face definitions improves proof accuracy.

Ensures the validity of previous results with revised arguments.

Abstract

Borodin et al figured out a gap of the paper published at J. Combinatorial Theory Ser. B (Vol.96 (2006) 958--963), and gave a new proof with the similar technique. The purpose of this note is to fix the gap by slightly revising the definition of special faces, and adding a few lines of explanation in the proofs (new added text are all in black font).