A Variation of the Erdős–Sós Conjecture in Bipartite Graphs

Abstract

The Erdős-Sós Conjecture states that every graph with average degree more than k − 2 contains all trees of order k as subgraphs. In this paper, we consider a variation of the above conjecture: studying the maximum size of an (n, m)-bipartite graph which does not contain all (k, l)-bipartite trees for given integers n ≥ m and k ≥ l. In particular, we determine that the maximum size of an (n, m)-bipartite graph which does not contain all (n, m)-bipartite trees as subgraphs (or all (k, 2)-bipartite trees as subgraphs, respectively). Furthermore, all these extremal graphs are characterized. B n,m , is a bipartite graph of order m + n whose vertices can be divided into two disjoint sets U and V with |U | = n and |V | = m such that every edge joins one vertex in U to another vertex in V . Moreover, denote by K n,m the complete (n, m)-bipartite graph. Furthermore, denote by B n,m the set of all bipartite graphs B n,m , and T n,m the set of all the (n, m)-bipartite trees T n,m with two partitions |U | = n and |V | = m, respectively.We call a problem a T urán type extremal problem if a family L of graphs are given from universe, such as G n is a graph of order n, we try to maximize e(G n ) of G n under the condition that G n does not contain L ∈ L. The maximum value is denoted by ex(n, L). Similarly, for a given family

Kalai's conjecture in $r$-partite $r$-graphs

“…Yuan and Zhang [16] conjecture similar results as (11) hold for all trees T (the exact bounds depend on how the bipartition sizes of T relate to n and m), and establish several special cases. Analogous improvements might be possible for hypergraphs.…”