Dynamic coloring parameters for graphs with given genus

• Published in: Discrete Applied Mathematics Vol. 235; pp. 129 - 141
• Main Authors: Loeb, Sarah, Mahoney, Thomas, Reiniger, Benjamin, Wise, Jennifer
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 30.01.2018; Elsevier BV
• Subjects: [formula omitted]-dynamic coloring; [formula omitted]-dynamic paintability; Graph coloring; Graph genus; Graphs; Paintability; Studies; Upper bounds; r-dynamic coloring; r-dynamic paintability; Graph genus
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2017.09.013

A proper vertex coloring of a graph G is r-dynamic if for each v∈V(G), at least min{r,d(v)} colors appear in NG(v). In this paper we investigate r-dynamic versions of coloring, list coloring, and paintability. We prove that planar and toroidal graphs are 3-dynamically 10-colorable, and this bound is sharp for toroidal graphs. We also give bounds on the minimum number of colors needed for any r in terms of the genus of the graph: for sufficiently large r, every graph with genus g is r-dynamically ((r+1)(g+5)+3)-colorable when g≤2 and r-dynamically ((r+1)(2g+2)+3)-colorable when g≥3. Furthermore, each of these upper bounds for r-dynamic k-colorability also holds for r-dynamic k-choosability and for r-dynamic k-paintability.