A note on the transversal size of a series of families constructed over Cycle Graph
Kaushik Majumder, Satyaki Mukherjee

TL;DR
This paper investigates the transversal size of specific intersecting families built over cycle graphs, providing an example with a number of blocks exceeding previous known bounds, thus contributing to extremal combinatorics.
Contribution
It introduces a new series of uniform intersecting families over cycle graphs with more blocks than previously known, challenging existing conjectures.
Findings
Constructed a series of families with over $(k/2)^{k-1}$ blocks
Computed the transversal size of these families
Provided a counterexample to earlier conjectures about maximal families
Abstract
Paul Erd\H{o}s and L\'{a}szl\'{o} Lov\'{a}sz established by means of an example that there exists a maximal intersecting family of sets with approximately blocks. L\'{a}szl\'{o} Lov\'{a}sz conjectured that their example is best known example which has the maximum number of blocks. Later it was disproved. But the quest for such examples remain valid till this date. In this short note, by computing transversal size of a certain series of uniform intersecting families constructed over the cycle graph, we provide an example which has more than (approximately) blocks.
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
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · graph theory and CDMA systems
