dc.contributor.author | Chudnovsky, Maria | |
dc.contributor.author | Fox, Jacob | |
dc.contributor.author | Scott, Alex | |
dc.contributor.author | Seymour, Paul | |
dc.contributor.author | Spirkl, Sophie | |
dc.date.accessioned | 2022-08-12 00:25:33 (GMT) | |
dc.date.available | 2022-08-12 00:25:33 (GMT) | |
dc.date.issued | 2020-11 | |
dc.identifier.uri | https://doi.org/10.1002/jgt.22556 | |
dc.identifier.uri | http://hdl.handle.net/10012/18509 | |
dc.description | This is the peer reviewed version of the following article: Chudnovsky, M., Fox, J., Scott, A., Seymour, P., & Spirkl, S. (2020). Pure pairs. III. Sparse graphs with no polynomial-sized anticomplete pairs. Journal of Graph Theory, 95(3), 315–340. https://doi.org/10.1002/jgt.22556, which has been published in final form at https://doi.org/10.1002/jgt.22556. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. | en |
dc.description.abstract | A graph is H-free if it has no induced subgraph isomorphic to H, and |G| denotes the number
of vertices of G. A conjecture of Conlon, Sudakov and the second author asserts that:
- For every graph H, there exists ∈ > 0 such that in every H-free graph G with |G| > 1 there
are two disjoint sets of vertices, of sizes at least ∈|G|∈ and ∈|G|, complete or anticomplete to
each other.
This is equivalent to:
- The "sparse linear conjecture": For every graph H, there exists ∈ > 0 such that in every
H-free graph G with |G| > 1, either some vertex has degree at least ∈|G|, or there are two
disjoint sets of vertices, of sizes at least ∈|G|∈ and ∈|G|, anticomplete to each other.
We prove a number of partial results towards the sparse linear conjecture. In particular, we prove
it holds for a large class of graphs H, and we prove that something like it holds for all graphs H.
More exactly, say H is "almost-bipartite" if H is triangle-free and V (H) can be partitioned into
a stable set and a set inducing a graph of maximum degree at most one. (This includes all graphs
that arise from another graph by subdividing every edge at least once.) Our main result is:
- The sparse linear conjecture holds for all almost-bipartite graphs H.
(It remains open when H is the triangle K3.) There is also a stronger theorem:
- For every almost-bipartite graph H, there exist ∈; t > 0 such that for every graph G with
|G| > 1 and maximum degree less than ∈|G|, and for every c with 0 < c ≤ 1, either G
contains ∈ct|G||H| induced copies of H, or there are two disjoint sets A,B V (G) with
|A| ≥ ∈ct|G| and |B| ≥ ∈|G| and with at most c|A| . |B| edges between them.
We also prove some variations on the sparse linear conjecture, such as:
- For every graph H, there exists ∈ > 0 such that in every H-free graph G with |G| > 1
vertices, either some vertex has degree at least ∈|G|, or there are two disjoint sets A, B of
vertices with |A| . |B| ≥ ∈|G|1+∈, anticomplete to each other. | en |
dc.description.sponsorship | Supported by NSF grant DMS-1550991. This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under grant number W911NF-16-1-0404. Supported by a Leverhulme Trust Research Fellowship Supported by ONR grant N00014-14-1-0084 and NSF grant DMS-1265563. | en |
dc.language.iso | en | en |
dc.publisher | Wiley | en |
dc.rights | Attribution-NonCommercial-NoDerivatives 4.0 International | * |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | * |
dc.subject | sparse linear conjecture | en |
dc.subject | pure pairs | en |
dc.title | Pure pairs. III. Sparse graphs with no polynomial-sized anticomplete pairs | en |
dc.type | Article | en |
dcterms.bibliographicCitation | Chudnovsky, M., Fox, J., Scott, A., Seymour, P., & Spirkl, S. (2020). Pure pairs. III. Sparse graphs with no polynomial-sized anticomplete pairs. Journal of Graph Theory, 95(3), 315–340. https://doi.org/10.1002/jgt.22556 | en |
uws.contributor.affiliation1 | Faculty of Mathematics | en |
uws.contributor.affiliation2 | Combinatorics and Optimization | en |
uws.typeOfResource | Text | en |
uws.peerReviewStatus | Reviewed | en |
uws.scholarLevel | Faculty | en |