dc.contributor.author | Chudnovsky, Maria | |
dc.contributor.author | Hompe, Patrick | |
dc.contributor.author | Scott, Alex | |
dc.contributor.author | Seymour, Paul | |
dc.contributor.author | Spirkl, Sophie | |
dc.date.accessioned | 2022-08-15 16:52:35 (GMT) | |
dc.date.available | 2022-08-15 16:52:35 (GMT) | |
dc.date.issued | 2022 | |
dc.identifier.uri | https://doi.org/10.37236/8451 | |
dc.identifier.uri | http://hdl.handle.net/10012/18546 | |
dc.description.abstract | Let x, y E (0, 1], and let A, B, C be disjoint nonempty stable subsets of a graph G, where every vertex in A has at least x |B| neighbors in B, and every vertex in B has at least y|C| neighbors in C, and there are no edges between A, C. We denote by ϕ(x, y) the maximum z such that, in all such graphs G, there is a vertex v E C that is joined to at least z|A| vertices in A by two-edge paths. This function has some interesting properties: we show, for instance, that ϕ (x, y) = ϕ (y, x) for all x, y, and there is a discontinuity in ϕ(x, x) where 1/x is an integer. For z= 1/2, 2/3, 1/3, 3/4, 2/5, 3/5, we try to find the (complicated) boundary between the set of pairs (x, y) with ϕ (x, y) ≥ z and the pairs with ϕ (x, y) < z. We also consider what happens if in addition every vertex in B has at least x |A| neighbors in A, and every vertex in C has at least y |B| neighbors in B.
We raise several questions and conjectures; for instance, it is open whether (x, x) ≥ 1/2 for all x > 1/3. | en |
dc.description.sponsorship | Supported by NSF grant DMS 1763817 and US Army Research Office Grant W911NF-16-1-0404. Supported by a Leverhulme Trust Research Fellowship. ‡Supported by ONR grant N00014-14-1-0084, AFOSR grant A9550-19-1-0187, and NSF grants DMS1265563 and DMS-1800053. This material is based upon work supported by the National Science Foundation under Award No. DMS-1802201. | en |
dc.language.iso | en | en |
dc.publisher | The Electronic Journal of Combinatorics | en |
dc.rights | Attribution-NoDerivatives 4.0 International | * |
dc.rights.uri | http://creativecommons.org/licenses/by-nd/4.0/ | * |
dc.subject | bipartite graphs | en |
dc.title | Concatenating Bipartite Graphs | en |
dc.type | Article | en |
dcterms.bibliographicCitation | Chudnovsky, M., Hompe, P., Scott, A., Seymour, P., & Spirkl, S. (2022). Concatenating Bipartite Graphs. The Electronic Journal of Combinatorics, P2.47-P2.47. https://doi.org/10.37236/8451 | 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 |