Extremal Graphs without Large Forbidden Subgraphs

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Abstract

The theory of extremal graphs without a fixed set of forbidden subgraphs is well developed. However, rather little is known about extremal graphs without forbidden subgraphs whose orders tend to ∞ with the order of the graph. In this note we deal with three problems of this latter type. Let L be a fixed bipartite graph and let L + Em be the join of L with the empty graph of order m. As our first problem we investigate the maximum of the size e(Gn) of a graph Gn (i.e. a graph of order n) provided Gn⊅L + E[cn, where c > 0 is a constant. In our second problem we study the maximum of e(Gn) if G n⊅K2(r,cn) and Gn⊅ K3 . The third problem is of a slightly different nature. Let Ck(t) be obtained from a cycle Ck by multiplying each vertex by t. We shall prove that if c > 0 then there exists a constant l(c) such that if Gn⊅Ck(t) for k = 3, 5, 2l(c) + 1, then one can omit [cn2] edges from G nso that the obtained graph is bipartite, provided n > n0 (c, t).

Original languageEnglish
Pages (from-to)29-41
Number of pages13
JournalAnnals of Discrete Mathematics
Volume3
Issue numberC
DOIs
Publication statusPublished - 1978

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Forbidden Subgraph
Extremal Graphs
Graph in graph theory
Bipartite Graph
Join
Tend
Cycle
Vertex of a graph

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

Cite this

Extremal Graphs without Large Forbidden Subgraphs. / Bollobás, B.; Erdős, P.; Simonovits, M.; Szemerédi, E.

In: Annals of Discrete Mathematics, Vol. 3, No. C, 1978, p. 29-41.

Research output: Contribution to journalArticle

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