List injective coloring of planar graphs with girth g≥6

A vertex coloring of a graph G is called injective if any two vertices with a common neighbor receive distinct colors. A graph G is injectively k -choosable if any list L of admissible colors on V ( G ) of size k allows an injective coloring ź such that ź ( v ) ź L ( v ) whenever v ź V ( G ) . The least k for which G is injectively k -choosable is denoted by ź i l ( G ) . In this paper, we show that if G is a planar graph with girth g ź 6 , then ź i l ( G ) ź Δ + 3 .

[1]  André Raspaud,et al.  Injective coloring of planar graphs , 2009, Discret. Appl. Math..

[2]  Wei Dong,et al.  Injective coloring of planar graphs with girth 6 , 2013, Discret. Math..

[3]  Daniel W. Cranston,et al.  Injective colorings of sparse graphs , 2010, Discret. Math..

[4]  Oleg V. Borodin,et al.  List 2-distance (Δ + 2)-coloring of planar graphs with girth 6 and Δ ≥ 24 , 2009 .

[5]  Oleg V. Borodin,et al.  Injective (Δ + 1)-coloring of planar graphs with girth 6 , 2011 .

[6]  Oleg V. Borodin,et al.  Sufficient conditions for the minimum 2-distance colorability of plane graphs of girth 6. , 2006 .

[7]  Daniel Král,et al.  Coloring squares of planar graphs with girth six , 2008, Eur. J. Comb..

[8]  Rui Li,et al.  Injective choosability of planar graphs of girth five and six , 2012, Discret. Math..

[9]  Mohammad R. Salavatipour,et al.  Frequency Channel Assignment on Planar Networks , 2002, ESA.

[10]  Oleg V. Borodin,et al.  2-distance (Delta+2)-coloring of planar graphs with girth six and Delta>=18 , 2009, Discret. Math..

[11]  Anna O. Ivanova,et al.  List 2-distance (Δ + 1)-coloring of planar graphs with girth at least 7 , 2011 .

[12]  André Raspaud,et al.  Some bounds on the injective chromatic number of graphs , 2010, Discret. Math..

[13]  Jan Kratochvíl,et al.  On the injective chromatic number of graphs , 2002, Discret. Math..

[14]  Mohammad R. Salavatipour,et al.  A bound on the chromatic number of the square of a planar graph , 2005, J. Comb. Theory, Ser. B.

[15]  Kai Lu,et al.  List injective coloring of planar graphs with girth 5, 6, 8 , 2013, Discret. Appl. Math..

[16]  Sheng Yang,et al.  List Injective Coloring of Planar graphs with girth G ≥ 5 , 2014, Discret. Math. Algorithms Appl..

[17]  Oleg V. Borodin,et al.  List 2-distance (Delta+2)-coloring of planar graphs with girth six , 2009, Eur. J. Comb..

[18]  Martin Tancer,et al.  Injective colorings of planar graphs with few colors , 2009, Discret. Math..

[19]  Oleg V. Borodin,et al.  List injective colorings of planar graphs , 2011, Discret. Math..

[20]  Magnús M. Halldórsson,et al.  Coloring powers of planar graphs , 2000, SODA '00.

[21]  G. Wegner.  Graphs with given diameter and a coloring problem , 1977 .

[22]  Daniel W. Cranston,et al.  Injective Colorings of Graphs with Low Average Degree , 2010, Algorithmica.

[23]  Kai Lu,et al.  Injective Coloring of Planar Graphs with girth 7 , 2012, Discret. Math. Algorithms Appl..