# The Game Coloring Number of Planar Graphs with a Specific Girth

The Game Coloring Number of Planar Graphs with a Specific Girth Let $$\mathrm{col_g}(G)$$ col g ( G ) be the game coloring number of a given graph G. Define the game coloring number of a family of graphs $$\mathcal {H}$$ H as $$\mathrm{col_g}(\mathcal {H}) := \max \{\mathrm{col_g}(G):G \in \mathcal {H}\}.$$ col g ( H ) : = max { col g ( G ) : G ∈ H } . Let $$\mathcal {P}_k$$ P k be the family of planar graphs of girth at least k. We show that $$\mathrm{col_g}(\mathcal {P}_7) \le 5.$$ col g ( P 7 ) ≤ 5 . This result extends a result about the game coloring number by Wang and Zhang [10] ( $$\mathrm{col_g}(\mathcal {P}_8) \le 5).$$ col g ( P 8 ) ≤ 5 ) . We also show that these bounds are sharp by constructing a graph G where $$G \in \mathcal {P}_k$$ G ∈ P k for each $$k \le 8$$ k ≤ 8 such that $$\mathrm{col_g}(G)=5.$$ col g ( G ) = 5 . As a consequence, $$\mathrm{col_g}(\mathcal {P}_k)=5$$ col g ( P k ) = 5 for $$k =7,8.$$ k = 7 , 8 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Graphs and Combinatorics Springer Journals

# The Game Coloring Number of Planar Graphs with a Specific Girth

, Volume 34 (2) – Feb 22, 2018
6 pages

/lp/springer_journal/the-game-coloring-number-of-planar-graphs-with-a-specific-girth-ZN1eJ8zHFo
Publisher
Springer Japan
Subject
Mathematics; Combinatorics; Engineering Design
ISSN
0911-0119
eISSN
1435-5914
D.O.I.
10.1007/s00373-018-1877-9
Publisher site
See Article on Publisher Site

### Abstract

Let $$\mathrm{col_g}(G)$$ col g ( G ) be the game coloring number of a given graph G. Define the game coloring number of a family of graphs $$\mathcal {H}$$ H as $$\mathrm{col_g}(\mathcal {H}) := \max \{\mathrm{col_g}(G):G \in \mathcal {H}\}.$$ col g ( H ) : = max { col g ( G ) : G ∈ H } . Let $$\mathcal {P}_k$$ P k be the family of planar graphs of girth at least k. We show that $$\mathrm{col_g}(\mathcal {P}_7) \le 5.$$ col g ( P 7 ) ≤ 5 . This result extends a result about the game coloring number by Wang and Zhang [10] ( $$\mathrm{col_g}(\mathcal {P}_8) \le 5).$$ col g ( P 8 ) ≤ 5 ) . We also show that these bounds are sharp by constructing a graph G where $$G \in \mathcal {P}_k$$ G ∈ P k for each $$k \le 8$$ k ≤ 8 such that $$\mathrm{col_g}(G)=5.$$ col g ( G ) = 5 . As a consequence, $$\mathrm{col_g}(\mathcal {P}_k)=5$$ col g ( P k ) = 5 for $$k =7,8.$$ k = 7 , 8 .

### Journal

Graphs and CombinatoricsSpringer Journals

Published: Feb 22, 2018

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