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Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs

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Title
Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs
Author(s)
Maria Axenovich; Casey Tompkins; Lea Weber
Publication Date
2021-05
Journal
JOURNAL OF GRAPH THEORY, v.97, no.1, pp.34 - 46
Publisher
WILEY
Abstract
For a bipartite graph G, let h similar to ( G ) be the largest t such that either G contains K t , t, a complete bipartite subgraph with parts of size t, or the bipartite complement of G contains K t , t as a subgraph. For a class of graphs F, let h similar to ( F ) = min { h similar to ( G ) : G is an element of F }. We say that a bipartite graph H is strongly acyclic if neither H nor its bipartite complement contains a cycle. By Forb ( n , H ) we denote the set of bipartite graphs with parts of size n, which do not contain H as an induced bipartite subgraph respecting the sides. One can easily show that h similar to ( Forb ( n , H ) ) = O ( n 1 - epsilon ) for a positive epsilon if H is not strongly acyclic. Here we ask whether h similar to ( Forb ( n , H ) ) is linear in n for any strongly acyclic graph H. We answer this question in the positive for all but four strongly acyclic graphs. We do not address this question for the remaining four graphs in this paper.
URI
https://pr.ibs.re.kr/handle/8788114/10036
DOI
10.1002/jgt.22639
ISSN
0364-9024
Appears in Collections:
Pioneer Research Center for Mathematical and Computational Sciences(수리 및 계산과학 연구단) > Discrete Mathematics Group(이산 수학 그룹) > 1. Journal Papers (저널논문)
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