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Books in graphs
, 2008
"... A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) n 2 wi ..."
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A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) n 2 with 0 < α < 17 −3 (), and G has no book of size at least graph G1 of order at least
Books versus Triangles
, 2009
"... A book of size b in a graph is an edge that lies in b triangles. Consider a graph G with n vertices and ⌊n 2 /4 ⌋ + 1 edges. Rademacher proved that G contains at least ⌊n/2 ⌋ triangles, and Erdős conjectured and Edwards proved that G contains a book of size at least n/6. We prove the following “line ..."
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A book of size b in a graph is an edge that lies in b triangles. Consider a graph G with n vertices and ⌊n 2 /4 ⌋ + 1 edges. Rademacher proved that G contains at least ⌊n/2 ⌋ triangles, and Erdős conjectured and Edwards proved that G contains a book of size at least n/6. We prove the following “linear combination ” of these two results. Suppose that α ∈ (1/2, 1) and the maximum size of a book in G is less than αn/2. Then G contains at least α(1 − α) n2 4 − o(n2) triangles as n → ∞. This is asymptotically sharp. On the other hand, for every α ∈ (1/3, 1/2), there exists β> 0 such that G contains at least βn 3 triangles. It remains an open problem to determine the largest possible β in terms of α. Our proof uses the RuzsaSzemerédi theorem. 1