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Doctoral Thesis
DOI
10.11606/T.45.2019.tde-15032019-114236
Document
Author
Full name
Luis Eduardo Zambrano Fernandez
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2018
Supervisor
Committee
Kohayakawa, Yoshiharu (President)
Benevides, Fabricio Siqueira
Hoppen, Carlos
Martin, Daniel Morgato
Mota, Guilherme Oliveira
Title in Portuguese
Densidade local em grafos
Keywords in Portuguese
Bisecções de grafos
Densidade local
Metades esparsas
Teorema de Turán
Abstract in Portuguese
Nós consideramos o seguinte problema. Fixado um grafo H e um número real \alpha \in (0,1], determine o menor \beta = \beta(\alpha, H) que satisfaz a seguinte propriedade: se G é um grafo de ordem n no qual cada subconjunto de [\alpha n] vértices induz mais que \beta n^2 arestas então G contém H como subgrafo. Este problema foi iniciado e motivado por Erdös ao conjecturar que todo grafo livre de triângulo de ordem n contém um subconjunto de [n/2] vértices que induz no máximo n^2 /50 arestas. Nosso resultado principal mostra que i) todo grafo de ordem n livre de triângulos e pentágonos contém um subconjunto de [n/2] vértices que induz no máximo n^2 /64 arestas, e ii) se G é um grafo regular de ordem n livre de triângulo, com grau excedendo n/3, então G contém um subconjunto de [n/2] vértices que induz no máximo n^2 /50 arestas. Se além disso G não é 3-cromático então G contém um subconjunto de [n/2] vértices que induz menos de n^2 /54 arestas. Como subproduto e confirmando uma conjectura de Erdös assintoticamente, temos que todo grafo regular de ordem n livre de triângulo com grau excedendo n/3 pode ser tornado bipartido pela omissão de no máximo (1/25 + o(1))n^2 arestas. Nós também fornecemos um contraexemplo a uma conjectura de Erdös, Faudree, Rousseau e Schelp.
Title in English
Local density in graphs
Keywords in English
Bisections of graphs
Local density
Sparse-halves
Turán's theorem
Abstract in English
We consider the following problem. Fixed a graph H and a real number \alpha \in (0,1], determine the smallest \beta = \beta(\alpha, H) satisfying the following property: if G is a graph of order n such that every subset of [\alpha n] vertices spans more that \beta n^2 edges then G contains H as a subgraph. This problem was initiated and motivated by Erdös who conjectured that every triangle-free graph of order n contains a subset of [n/2] vertices that spans at most n^2 /50 edges. Our main result shows that i) every triangle- and pentagon-free graph of order n contains a subset of [n/2] vertices inducing at most n^2 /64 edges and, ii) if G is a triangle-free regular graph of order n with degree exceeding n/3 then G contains a subset of [n/2] vertices inducing at most n^2 /50 edges. Furthermore, if G is not 3-chromatic then G contains a subset of [n/2] vertices inducing less than n^2 /54 edges. As a by-product and confirming a conjecture of Erdös asymptotically, we obtain that every n-vertex triangle-free regular graph with degree exceeding n/3 can be made bipartite by removing at most (1/25 + o(1))n^2 edges. We also provide a counterexample to a conjecture of Erdös, Faudree, Rousseau and Schelp.
 
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teseZambrano.pdf (1.26 Mbytes)
Publishing Date
2019-03-22
 
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