Acyclic chromatic index of planar graphs with triangles

Abstract A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ a ′ ( G ) , is the least number of colors in an acyclic edge coloring of G. Let G be a planar graph with maximum degree Δ ( G ) . In this paper, we show that χ a ′ ( G ) ⩽ Δ ( G ) + 4 , if G contains no 4-cycle; χ a ′ ( G ) ⩽ Δ ( G ) + 5 , if G contains no intersecting triangles; and χ a ′ ( G ) ⩽ Δ ( G ) + 6 if G contains no adjacent triangles.

[1]  Jozef Fiamčík.  Acyclic chromatic index of a subdivided graph , 1984 .

[2]  Bin Liu,et al.  Acyclic edge chromatic number of outerplanar graphs , 2010 .

[3]  Mariusz Haluszczak,et al.  About acyclic edge colourings of planar graphs , 2008, Inf. Process. Lett..

[4]  C. R. Subramanian,et al.  Acyclic Edge Colouring of Outerplanar Graphs , 2007, AAIM.

[5]  Jozef Fiamčík,et al.  Acyclic chromatic index of a graph with maximum valency three , 1980 .

[6]  Noga Alon,et al.  A Parallel Algorithmic Version of the Local Lemma , 1991, Random Struct. Algorithms.

[7]  Manu Basavaraju,et al.  Acyclic Edge Coloring of Triangle-Free Planar Graphs , 2012, J. Graph Theory.

[8]  Noga Alon,et al.  Algorithmic Aspects of Acyclic Edge Colorings , 2001, Algorithmica.

[9]  C. R. Subramanian,et al.  Optimal Acyclic Edge Colouring of Grid Like Graphs , 2006, COCOON.

[10]  Roman Soták,et al.  Acyclic edge coloring of planar graphs with Δ colors , 2012, Discret. Appl. Math..

[11]  Manu Basavaraju,et al.  Acyclic edge coloring of graphs with maximum degree 4 , 2009 .

[12]  Bin Liu,et al.  Acyclic edge colorings of planar graphs and seriesparallel graphs , 2009 .

[13]  Noga Alon,et al.  Acyclic edge colorings of graphs , 2001 .

[14]  San Skulrattanakulchai,et al.  Acyclic colorings of subcubic graphs , 2004, Inf. Process. Lett..

[15]  Jianfeng Hou,et al.  Improved bounds for acyclic chromatic index of planar graphs , 2011, Discret. Appl. Math..

[16]  Manu Basavaraju,et al.  Acyclic edge coloring of subcubic graphs , 2008, Discret. Math..

[17]  Bruce A. Reed,et al.  Further algorithmic aspects of the local lemma , 1998, STOC '98.

[18]  Mieczyslaw Borowiecki,et al.  Acyclic edge colouring of planar graphs without short cycles , 2010, Discret. Math..