English

Acyclic edge coloring conjecture is true on planar graphs without intersecting triangles

Discrete Mathematics 2020-05-14 v1 Combinatorics

Abstract

An acyclic edge coloring of a graph GG is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiam{\v{c}}ik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph with maximum degree Δ\Delta is acyclically edge (Δ+2)(\Delta + 2)-colorable. Despite many milestones, the conjecture remains open even for planar graphs. In this paper, we confirm affirmatively the conjecture on planar graphs without intersecting triangles. We do so by first showing, by discharging methods, that every planar graph without intersecting triangles must have at least one of the six specified groups of local structures, and then proving the conjecture by recoloring certain edges in each such local structure and by induction on the number of edges in the graph.

Keywords

Cite

@article{arxiv.2005.06152,
  title  = {Acyclic edge coloring conjecture is true on planar graphs without intersecting triangles},
  author = {Qiaojun Shu and Guohui Lin and Eiji Miyano},
  journal= {arXiv preprint arXiv:2005.06152},
  year   = {2020}
}

Comments

Main body 23 pages, with 23 more pages of detailed proof in the Appendix; an extended abstract appears in Proceedings of TAMC 2020

R2 v1 2026-06-23T15:30:23.640Z