Abstract
We consider a collection of coloring games played on finite graphs. These games all involve two players, Alice and Bob, alternating coloring the vertices (or edges or both) from a finite set of colors. At each step, the players must use legal colors, where the definition of legal depends on which variation of the game is being considered. The first player, Alice, wants to ensure that all relevant elements (vertices and/or edges) are colored. The second player, Bob, wants to force a situation in which there is an uncolored element that cannot be legally colored. For each version of the game the parameter of interest finds the least number of colors that must be available so that Alice has a winning strategy for the game. We examine this game for a number of classes of graphs, including trees, forests, outerplanar graphs, and planar graphs. We also note a number of unexpected interesting properties these parameters have and highlight how rich these games can be to explore for undergraduate researchers and their mentors.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Andres, S.D.: The game chromatic index of forests of maximum degree Δ ≥ 5. Discret. Appl. Math. 154(9), 1317–1323 (2006)
Appel, K., Haken, W., Koch, J.: Every planar map is four colorable. Part II: reducibility. Ill. J. Math. 21, 491–567 (1977)
Bartnicki, T., Grytczuk, J., Kierstead, H.A., Zhu, X.: The map-coloring game. Am. Math. Mon. 114(9), 793–803 (2007)
Bodlaender, H.: On the complexity of some coloring games. In: Möhring, R. (ed.) Graph Theoretical Concepts in Computer Science, vol. 484, pp. 30–40. Lecture notes in Computer Science. Springer, Berlin (1991)
Cai, L., Zhu, X.: Game chromatic index of k-degenerate graphs. J. Graph Theory 36(3), 144–155 (2001)
Chou, C., Wang, W., Zhu, X.: Relaxed game chromatic number of graphs. Discret. Math. 262(1–3), 89–98 (2003)
Cohen-Addad, V., Hebdige M., Král’, D., Li, Z., Salgado, E.: Steinberg’s conjecture is false. Preprint arXiv:1604.05108 (2016)
Cowen, L., Cowen, R., Woodall, D.: Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency. J. Graph Theory 10, 187–195 (1986)
Cowen, L., Goddard, W., Jesurum, C.: Defective coloring revisited. J. Graph Theory 24, 205–219 (1997)
Deuber, W., Zhu, X.: Relaxed coloring of a graph. Graphs Combin. 14, 121–130 (1998)
Dinski, T., Zhu, X.: A bound for the game chromatic number of graphs. Discret. Math. 196, 109–115 (1999)
Dunn, C.: Complete multipartite graphs and the relaxed coloring game. Order 29(3), 507–512 (2012)
Dunn, C.: The relaxed game chromatic index of k-degenerate graphs. Discret. Math. 307, 1767–1775 (2007)
Dunn, C., Kierstead, H.A.: A simple competitive graph coloring algorithm II. J. Combin. Theory Series B, 90, 93–106 (2004)
Dunn, C., Kierstead, H.A.: A simple competitive graph coloring algorithm III. J. Combin. Theory Ser. B 92, 137–150 (2004)
Dunn, C., Kierstead, H.A.: The relaxed game chromatic number of outerplanar graphs. J. Graph Theory 46, 69–78 (2004)
Dunn, C., Naymie, C., Nordstrom, J.F., Pitney, E., Sehorn, W., Suer, C.: Clique-relaxed graph coloring. Involve 4(2), 127–138 (2011)
Dunn, C., Larsen, V., Lindke, K., Retter, T., Toci, D.: The game chromatic number of trees and forests. Discret. Math. Theor. Comput. Sci. 17(2), 31–48 (2015)
Dunn, C., Morawski, D., Nordstrom, J.F.: The relaxed edge-coloring game and k-degenerate graphs. Order 32(3), 347–361 (2015)
Dunn, C., Hays, T., Naftz, L., Nordstrom, J.F., Samelson, E., Vega, J., Total coloring games (in preparation)
Eaton, N., Hull, T.: Defective list colorings of planar graphs. Bull. Inst. Combin. Appl. 25, 79–87 (1999)
Erdős, P., Faigle, U., Hochstättler, W., Kern, W.: Note on the game chromatic index of trees. Theor. Comput. Sci. 313(3), 371–376 (2004)
Faigle, U., Kern, W., Kierstead, H.A., Trotter, W.: On the game chromatic number of some classes of graphs. Ars Combinatoria 35, 143–150 (1993)
Gardner, M.: Mathematical games. Sci. Am. 23, (1981)
Guan, D., Xhu, X.: Game chromatic number of outerplanar graphs. J. Graph Theory 30, 67–70 (1999)
He, W., Wu, J., Zhu, X.: Relaxed game chromatic number of trees and outerplanar graphs. Discret. Math. 281(1–3), 209–219 (2004)
Kierstead, H.A.: A simple competitive graph coloring algorithm. J. Combin. Theory Ser. B 78, 57–68 (2000)
Kierstead, H.A., Trotter, W.: Planar graph coloring with an uncooperative partner. J. Graph Theory 18, 569–584 (1994)
Lam, P., Shiu, W., Xu, B.: Edge game-coloring of graphs. Graph Theory Notes N. Y. XXXVII, 17–19 (1999)
Zhu, X.: The game coloring number of planar graphs. J. Combin. Theory Ser. B 75, 245–258 (1999)
Zhu, X.: The game coloring number of pseudo partial k-trees. Discret. Math. 215, 245–262 (2000)
Zhu, X.: Refined activation strategy for the marking game. J. Combin. Theory Ser. B 98, 1–18 (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Dunn, C., Larsen, V., Nordstrom, J.F. (2017). Introduction to Competitive Graph Coloring. In: Wootton, A., Peterson, V., Lee, C. (eds) A Primer for Undergraduate Research. Foundations for Undergraduate Research in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-66065-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-66065-3_5
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-66064-6
Online ISBN: 978-3-319-66065-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)