In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to...In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs (in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ0) be the set of graphs G with n vertices and minimum degree 50, and J(n, Δ) be the set of graphs G with n vertices and maximum degree A. We study the four following extremal problems on graphs: a(n,δ0) = min{δ(G) | G ∈H(n, δ0)}, b(n, δ0) =- max{δ(G)| e ∈H(n, δ0)}, α(n, Δ) = min{δ(G) [ G ∈ J(n, Δ)} and β(n,Δ) = max{δ(G) ] G∈Π(n,Δ)}. In particular, we obtain bounds for b(n, δ0) and we compute the precise value of a(n, δ0), α(n, Δ) and w(n, Δ) for all values of n, r0 and A, respectively.展开更多
The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake: in Theorem 6.2 appears that β(n, △) = (n-△ + 5)/4 but the correct statement is...The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake: in Theorem 6.2 appears that β(n, △) = (n-△ + 5)/4 but the correct statement is β(n, △) = (n -△ + 4)/4. In this erratum we correct the theorem and give the correct proof.展开更多
基金Supported in part by two grants from Ministerio de Economía y Competitividad,Spain:MTM2013-46374-P and MTM2015-69323-REDT
文摘In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs (in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ0) be the set of graphs G with n vertices and minimum degree 50, and J(n, Δ) be the set of graphs G with n vertices and maximum degree A. We study the four following extremal problems on graphs: a(n,δ0) = min{δ(G) | G ∈H(n, δ0)}, b(n, δ0) =- max{δ(G)| e ∈H(n, δ0)}, α(n, Δ) = min{δ(G) [ G ∈ J(n, Δ)} and β(n,Δ) = max{δ(G) ] G∈Π(n,Δ)}. In particular, we obtain bounds for b(n, δ0) and we compute the precise value of a(n, δ0), α(n, Δ) and w(n, Δ) for all values of n, r0 and A, respectively.
基金Supported by two grants from Ministerio de Economía y Competitividad,Spain(Grant Nos.MTM2013-46374-P and MTM2015-69323-REDT)
文摘The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake: in Theorem 6.2 appears that β(n, △) = (n-△ + 5)/4 but the correct statement is β(n, △) = (n -△ + 4)/4. In this erratum we correct the theorem and give the correct proof.