# On the Game Chromatic Number of Sparse Random Graphs

## Abstract

*Random Structures Algorithms*, 32 (2008), pp. 223--235] began the analysis of the asymptotic behavior of this parameter for a random graph $G_{n,p}$. This paper provides some further analysis for graphs with constant average degree, i.e., $np=O(1)$, and for random regular graphs. We show that with high probability (w.h.p.) $c_1\chi(G_{n,p})\leq \chi_g(G_{n,p})\leq c_2\chi(G_{n,p})$ for some absolute constants $1<c_1< c_2$. We also prove that if $G_{n,3}$ denotes a random $n$-vertex cubic graph, then w.h.p. $\chi_g(G_{n,3})=4$.

### Keywords

### MSC codes

## Get full access to this article

View all available purchase options and get full access to this article.

## References

*The chromatic number of random regular graphs*, in Proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 8th International Workshop on Randomization and Computation, Lecture Notes in Comput. Sci. 3122, Springer, New York, 2004, pp. 219--228.

*The two possible values of the chromatic number of a random graph*, Ann. of Math. (2), 162 (2005), pp. 1333--1349.

*The Probabilistic Method*, 3rd ed., John Wiley, New York, 2008.

*The map coloring game*, Amer. Math. Monthly, 114 (2007), pp. 793--803.

*On the complexity of some coloring games*, Internat. J. Found. Comput. Sci., 2 (1991), pp. 133--147.

*The game chromatic number of random graphs*, Random Structures Algorithms, 32 (2008), pp. 223--235.

*The chromatic number of random graphs*, Combinatorica, 8 (1988), pp. 49--55.

*A probabilistic proof of an asymptotic formula for the number of labelled regular graphs*, European J. Combin., 1 (1980), pp. 311--316.

*Random regular graphs of non-constant degree: Connectivity and Hamilton cycles*, Combin. Probab. Comput., 11 (2002), pp. 249--262.

*On the game chromatic number of some classes of graphs*, Ars Combin., 35 (1993), pp. 143--150.

*On the independence and chromatic numbers of random regular graphs*, J. Combin. Theory Ser. B, 54 (1992), pp. 123--132.

*Mathematical games*, Sci. Amer., 244 (1981), pp. 18--26.

*Random Graphs*, John Wiley, New York, 2000.

*On the chromatic number of random $d$-regular graphs*, Adv. Math., 223 (2010), pp. 300--328.

*Sandwiching random graphs: Universality between random graph models*, Adv. Math., 188 (2004), pp. 444--469.

*The chromatic number of random graphs*, Combinatorica, 11 (1991), pp. 45--54.

*Models of random regular graphs*, in Surveys in Combinatorics, J. D. Lamb and D. A. Preece, eds., London Math. Soc. Lecture Note Ser. 276, Cambridge University Press, Cambridge, UK, 1999, pp. 239--298.

## Information & Authors

### Information

#### Published In

#### Copyright

#### History

**Submitted**: 10 January 2012

**Accepted**: 10 December 2012

**Published online**: 18 April 2013

#### Keywords

#### MSC codes

### Authors

## Metrics & Citations

### Metrics

### Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

#### Cited By

- The eternal game chromatic number of random graphsEuropean Journal of Combinatorics, Vol. 95 | 1 Jun 2021
- The Game Chromatic Number of a Random HypergraphDiscrete Mathematics and Applications | 22 November 2020