2024/09/23 by Pokrovskiy, Alexey · 1 citation
#05C05 #05C35 #05C75 #Combinatorics (math.CO) #FOS: Mathematics #G.2.2
paper · doi:10.48550/arxiv.2409.15189
This is a companion paper to the paper "Hyperstability in the Erdos-Sos Conjecture". In that paper the following rough structure theorem was proved for graphs G containing no copy of a bounded degree tree T: from any such G, one can delete o(|G||T|) edges in order to get a subgraph all of whose connected components have a cover of order 3|T|. This theorem creates an incentive for studying graphs whose connected components have covers of order O(|T|) - and this is what will be explored here. It turns out that such graphs are amenable to regularity approaches which have been successful in studying dense T-free graphs. In this paper we will follow such an approach from the paper "On the Erdos-Sos conjecture for trees with bounded degree" by Besomi, Pavez-Signe, and Stein and show how it can be adapted from dense graphs to graphs with a small cover.