157,872 research outputs found

    Tur\'an Graphs, Stability Number, and Fibonacci Index

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    The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an graphs and a connected variant of them are also extremal for these particular problems.Comment: 11 pages, 3 figure

    New Computational Upper Bounds for Ramsey Numbers R(3,k)

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    Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <= 68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are improvements by one over the previously best known bounds. Let e(3,k,n) denote the minimum number of edges in any triangle-free graph on n vertices without independent sets of order k. The new upper bounds on R(3,k) are obtained by completing the computation of the exact values of e(3,k,n) for all n with k <= 9 and for all n <= 33 for k = 10, and by establishing new lower bounds on e(3,k,n) for most of the open cases for 10 <= k <= 15. The enumeration of all graphs witnessing the values of e(3,k,n) is completed for all cases with k <= 9. We prove that the known critical graph for R(3,9) on 35 vertices is unique up to isomorphism. For the case of R(3,10), first we establish that R(3,10) = 43 if and only if e(3,10,42) = 189, or equivalently, that if R(3,10) = 43 then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that R(3,10) <= 42.Comment: 28 pages (includes a lot of tables); added improved lower bound for R(3,11); added some note

    Supersaturation Problem for Color-Critical Graphs

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    The \emph{Tur\'an function} \ex(n,F) of a graph FF is the maximum number of edges in an FF-free graph with nn vertices. The classical results of Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs where the key question is to determine hF(n,q)h_F(n,q), the minimum number of copies of FF that a graph with nn vertices and \ex(n,F)+q edges can have. We determine hF(n,q)h_F(n,q) asymptotically when FF is \emph{color-critical} (that is, FF contains an edge whose deletion reduces its chromatic number) and q=o(n2)q=o(n^2). Determining the exact value of hF(n,q)h_F(n,q) seems rather difficult. For example, let c1c_1 be the limit superior of q/nq/n for which the extremal structures are obtained by adding some qq edges to a maximum FF-free graph. The problem of determining c1c_1 for cliques was a well-known question of Erd\H os that was solved only decades later by Lov\'asz and Simonovits. Here we prove that c1>0c_1>0 for every {color-critical}~FF. Our approach also allows us to determine c1c_1 for a number of graphs, including odd cycles, cliques with one edge removed, and complete bipartite graphs plus an edge.Comment: 27 pages, 2 figure
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