61 research outputs found

    Hamiltonicity, independence number, and pancyclicity

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    A graph on n vertices is called pancyclic if it contains a cycle of length l for all 3 \le l \le n. In 1972, Erdos proved that if G is a Hamiltonian graph on n > 4k^4 vertices with independence number k, then G is pancyclic. He then suggested that n = \Omega(k^2) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such that n > ck^{7/3} suffices

    Number of cliques in graphs with a forbidden subdivision

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    We prove that for all positive integers tt, every nn-vertex graph with no KtK_t-subdivision has at most 250tn2^{50t}n cliques. We also prove that asymptotically, such graphs contain at most 2(5+o(1))tn2^{(5+o(1))t}n cliques, where o(1)o(1) tends to zero as tt tends to infinity. This strongly answers a question of D. Wood asking if the number of cliques in nn-vertex graphs with no KtK_t-minor is at most 2ctn2^{ct}n for some constant cc.Comment: 10 pages; to appear in SIAM J. Discrete Mat

    Two Approaches to Sidorenko's Conjecture

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    Sidorenko's conjecture states that for every bipartite graph HH on {1,⋯ ,k}\{1,\cdots,k\}, ∫∏(i,j)∈E(H)h(xi,yj)dμ∣V(H)∣β‰₯(∫h(x,y) dΞΌ2)∣E(H)∣\int \prod_{(i,j)\in E(H)} h(x_i, y_j) d\mu^{|V(H)|} \ge \left( \int h(x,y) \,d\mu^2 \right)^{|E(H)|} holds, where ΞΌ\mu is the Lebesgue measure on [0,1][0,1] and hh is a bounded, non-negative, symmetric, measurable function on [0,1]2[0,1]^2. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph HH to a graph GG is asymptotically at least the expected number of homomorphisms from HH to the Erd\H{o}s-R\'{e}nyi random graph with the same expected edge density as GG. In this paper, we present two approaches to the conjecture. First, we introduce the notion of tree-arrangeability, where a bipartite graph HH with bipartition AβˆͺBA \cup B is tree-arrangeable if neighborhoods of vertices in AA have a certain tree-like structure. We show that Sidorenko's conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko's conjecture holds if there are two vertices a1,a2a_1, a_2 in AA such that each vertex a∈Aa \in A satisfies N(a)βŠ†N(a1)N(a) \subseteq N(a_1) or N(a)βŠ†N(a2)N(a) \subseteq N(a_2), and also implies a recent result of Conlon, Fox, and Sudakov \cite{CoFoSu}. Second, if TT is a tree and HH is a bipartite graph satisfying Sidorenko's conjecture, then it is shown that the Cartesian product Tβ–‘HT \Box H of TT and HH also satisfies Sidorenko's conjecture. This result implies that, for all dβ‰₯2d \ge 2, the dd-dimensional grid with arbitrary side lengths satisfies Sidorenko's conjecture.Comment: 20 pages, 2 figure

    Long paths and cycles in random subgraphs of graphs with large minimum degree

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    For a given finite graph GG of minimum degree at least kk, let GpG_{p} be a random subgraph of GG obtained by taking each edge independently with probability pp. We prove that (i) if pβ‰₯Ο‰/kp \ge \omega/k for a function Ο‰=Ο‰(k)\omega=\omega(k) that tends to infinity as kk does, then GpG_p asymptotically almost surely contains a cycle (and thus a path) of length at least (1βˆ’o(1))k(1-o(1))k, and (ii) if pβ‰₯(1+o(1))ln⁑k/kp \ge (1+o(1))\ln k/k, then GpG_p asymptotically almost surely contains a path of length at least kk. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking GG to be the complete graph on k+1k+1 vertices.Comment: 26 page
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