The Erdős–Sós Conjecture for Graphs withoutC4

Constructing trees in graphs with no K2,s

“…Brandt and the second author [2] showed the conjecture holds for graphs with girth at least 5. Saclé and Woźniak [9] then improved this result and showed the Erdős-Sós conjecture holds for graphs without a K 2,2 . Haxell [6] showed the conjecture holds for graphs that do not contain a K 2,s , where s = k/18 .…”

On the Loebl-Koml�s-S�s conjecture

“…As remarked in [5], the condition that the average degree of the graph G is greater than k À 1 from the Erdo ÂÂs±So Âs conjecture is replaced in Loebl±Komlo Âs± So Âs conjecture by the condition that the medium degree of G is greater than k (for some special cases of the Erdo ÂÂs±So Âs conjecture see for example [8] as well as [2] and [7]). …”

The Erdős‐Sós Conjecture for trees of diameter four

“…The condition that G has girth at least 5 means that it has neither 3-cycles nor 4-cycles. Saclé and Woźniak [7] show that it suffices to assume that G has no 4-cycles. (d) The complement of G is the graph G c :¼ ðV; E c Þ where E c :¼ fuv : u; v 2 V; uv = 2 Eg.…”