Induced Tur\'an numbers
Po-Shen Loh, Michael Tait, Craig Timmons, and Rodrigo Zhou

TL;DR
This paper extends Turán-type extremal results to graphs forbidding certain subgraphs both as induced and non-induced, providing new bounds and insights into the structure of such graphs.
Contribution
It proves that classical Turán bounds hold for graphs forbidding a fixed subgraph both as induced and non-induced, and derives bounds on clique counts in these graphs.
Findings
Classical Turán bounds extend to induced and non-induced forbidden subgraphs.
Derived bounds on the number of fixed-size cliques in $K_r$-free graphs with induced $K_{s,t}$-free condition.
Established a new angle for generalizing Turán theory to induced subgraph constraints.
Abstract
The classical K\H{o}v\'ari-S\'os-Tur\'an theorem states that if is an -vertex graph with no copy of as a subgraph, then the number of edges in is at most . We prove that if one forbids as an induced/ subgraph, and also forbids any/ fixed graph as a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a nontrivial angle from which to generalize Tur\'an theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a nontrivial upper bound on the number of cliques of fixed order in a -free graph with no induced copy of . This result is an induced analog of a recent theorem of Alon and Shikhelman and is of independent interest.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph theory and applications
