Abstract
A large body of research in graph theory concerns the induced subgraphs of graphs with large chromatic number, and especially which induced cycles must occur. In this paper, we unify and substantially extend results from a number of previous papers, showing that, for every positive integer k, every graph with large chromatic number contains either a large complete subgraph or induced cycles of all lengths modulo k. As an application, we prove two conjectures of Kalai and Meshulam from the 1990’s connecting the chromatic number of a graph with the homology of its independence complex.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Bonamy, P. Charbit and S. Thomassé: Graphs with large chromatic number induce 3k-cycles, submitted for publication, arXiv:1408.2172.
M. Chudnovsky, A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. II. Three steps towards Gyárfás' conjectures, J. Combinatorial Theory, Ser. B118 (2016), 109–128.
M. Chudnovsky, A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. III. Long holes, Combinatorica37 (2017), 1057–72.
M. Chudnovsky, A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. V. Chandeliers and strings, submitted for publication, arXiv:1609.00314.
M. Chudnovsky, A. Scott, P. Seymour and S. Spirkl: Induced subgraphs of graphs with large chromatic number. VIII. Long odd holes, submitted for publication, arXiv:1701.07217.
M. Chudnovsky, A. Scott, P. Seymour and S. Spirkl: Proof of the Kalai-Meshulam conjecture, submitted for publication, arXiv:1810.00065.
A. Engstrom: Graph colouring and the total Betti number, arXiv:1412.8460.
A. Gyárfás: Problems from the world surrounding perfect graphs, Proceedings of the International Conference on Combinatorial Analysis and its Applications, (Pokrzywna, 1985), Zastos. Mat.19 (1987), 413–441.
A. Hatcher: Algebraic Topology, Cambridge University Press (2009).
G. Kalai: When do a few colors suffice?, https://gilkalai.wordpress.com/2014/12/19/when-a-few-colors-suffice.
G. Kalai: Combinatorial expectations from commutative algebra, Oberwolfach Reports1 (2004), 1725–1730.
A. Scott: Induced cycles and chromatic number, J. Combinatorial Theory, Ser. B76 (1999), 150–154.
A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. I. Odd holes, J. Combinatorial Theory, Ser. B121 (2016), 68–84.
A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes, J. Combinatorial Theory, Ser. B132 (2018), 180–235.
A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. VI. Banana trees, submitted for publication, arXiv:1701.05597.
A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. VII. Gyárfás' complementation conjecture, submitted for publication, arXiv:1701.06301.
Acknowledgement
We would like to thank Gil Kalai for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by a Leverhulme Trust Research Fellowship.
Supported by ONR grant N00014-14-1-0084, NSF grant DMS-1265563, NSF grant DMS-1800053 and AFOSR grant A9550-19-1-0187.
Rights and permissions
About this article
Cite this article
Scott, A., Seymour, P. Induced Subgraphs of Graphs With Large Chromatic Number. X. Holes of Specific Residue. Combinatorica 39, 1105–1132 (2019). https://doi.org/10.1007/s00493-019-3804-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-019-3804-y