Counting extensions revisited. Issue 1 (25th February 2022)

• Record Type: Journal Article
• Authors: Šileikis, Matas; Warnke, Lutz
• Is Part Of: Random structures & algorithms. Volume 61:Issue 1(2022)
• Page Start: 3
• Page End: 30
• Publication Date: 2022-02-25
• Subjects: extreme values -- random graph -- rooted subgraphs -- subgraph counts; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.21050
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: We consider rooted subgraphs in random graphs, that is, extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. In 1989, Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices. For the important strictly balanced case, Spencer also raised the fundamental question as to whether these conditions are necessary. We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open.