Abstract
AbstractFor graphs and a family of graphs , we write to denote that every blue‐red coloring of the edges of contains either a blue copy of , or a red copy of each . For integers and , let denote the family of all trees with edges and maximum degree at most . We prove that for each , there exist constants such that if and , then with high probability. This is a random version of a well‐known result of Chvátal from 1977. The proof combines a stability argument with the embedding of trees in expander graphs. Furthermore, the proof of the stability result is based on a sparse random analogue of the Erdős–Sós conjecture for trees with linear size and bounded maximum degree, which may be of independent interest.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.