The Erdős‐Sós Conjecture for trees of diameter four

Abstract

The Erdős-Só s Conjecture is that a finite graph G with average degree greater than k À 2 contains every tree with k vertices. Theorem 1 is a special case: every k-vertex tree of diameter four can be embedded in G. A more technical result, Theorem 2, is obtained by extending the main ideas in the proof of Theorem 1. ß 2005 Wiley Periodicals, Inc. J Graph Theory 49: [291][292][293][294][295][296][297][298][299][300][301] 2005

The spectral radius of graphs without trees of diameter at most four

“…The Turán type version of Conjecture 1 is the well-known Erdős-Sós Conjecture [1] which states that every finite simple graph with average degree greater than k − 2 contains a copy of any tree of order k as a subgraph. The Erdős-Sós Conjecture attracts many attentions and was verified for many specific family of trees, especially for trees of diameter at most four [4].…”

A Variation of the Erdős–Sós Conjecture in Bipartite Graphs

“…In [1,2,3], Ajtai, Komlós, Simonovits and Szemerédi proved that the Erdős-Sós Conjecture is true for sufficiently large k. Fan [12] proved that the Erdős-Sós Conjecture holds for the spiders of large size. More results on this conjecture can be referred to [4,6,8,9,18,20,22,23,24,25,26]. On the extremal problems on complete bipartite graph, Kővári, Sós and Turán [17] proved the following result: Theorem 1.3 [17] The maximum size of a graph containing no complete bipartite graph K a,b is at most 1 2 a √ b − 1n 2−1/a + 1 2 (a − 1)n.…”

An Erdős-Gallai type theorem for vertex colored graphs

“…We may assume by induction that δ(G) > a+b 2 ≥ a. Since ex(m, S a,b ) = m a+b 2 (see, for example [6]), it follows that G contains a copy of S a,b . Suppose this copy is defined by the edge {u, v} together with the disjoint sets A ⊆ N (u), B ⊆ N (v) with |A|= a, |B|= b.…”