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Graphs and Combinatorics

, Volume 33, Issue 4, pp 859–868 | Cite as

Acyclic Chromatic Index of Triangle-free 1-Planar Graphs

  • Jijuan Chen
  • Tao WangEmail author
  • Huiqin Zhang
Original Paper

Abstract

An acyclic edge coloring of a graph G is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index \(\chi _{a}'(G)\) of a graph G is the least number of colors in an acyclic edge coloring of G. It was conjectured that \(\chi '_{a}(G)\le {\varDelta }(G) + 2\) for any simple graph G with maximum degree \({\varDelta }(G)\). A graph is 1-planar if it can be drawn on the plane such that every edge is crossed by at most one other edge. In this paper, we show that every triangle-free 1-planar graph G has an acyclic edge coloring with \({\varDelta }(G) + 16\) colors.

Keywords

Acyclic edge coloring Acyclic chromatic index \(\kappa \)-Deletion-minimal graph 1-Planar graph 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and careful reading of this paper.

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Copyright information

© Springer Japan 2017

Authors and Affiliations

  1. 1.Institute of Applied MathematicsHenan UniversityKaifengPeople’s Republic of China

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