The largest subgraph without a forbidden induced subgraph
arxiv(2024)
摘要
We initiate the systematic study of the following Turán-type question.
Suppose Γ is a graph with n vertices such that the edge density
between any pair of subsets of vertices of size at least t is at most 1 -
c, for some t and c > 0. What is the largest number of edges in a subgraph
G ⊆Γ which does not contain a fixed graph H as an induced
subgraph or, more generally, which belongs to a hereditary property
𝒫? This provides a common generalization of two recently studied
cases, namely Γ being a (pseudo-)random graph and a graph without a
large complete bipartite subgraph. We focus on the interesting case where H
is a bipartite graph.
We determine the answer up to a constant factor with respect to n and t,
for certain bipartite H and for Γ either a dense random graph or a
Paley graph with a square number of vertices. In particular, our bounds match
if H is a tree, or if one part of H has d vertices complete to the other
part, all other vertices in that part have degree at most d, and the other
part has sufficiently many vertices. As applications of the latter result, we
answer a question of Alon, Krivelevich, and Samotij on the largest subgraph
with a hereditary property which misses a bipartite graph, and determine up to
a constant factor the largest number of edges in a string subgraph of Γ.
The proofs are based on a variant of the dependent random choice and a novel
approach for finding induced copies by inductively defining probability
distributions supported on induced copies of smaller subgraphs.
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