396 research outputs found

    Unicyclic Components in Random Graphs

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    The distribution of unicyclic components in a random graph is obtained analytically. The number of unicyclic components of a given size approaches a self-similar form in the vicinity of the gelation transition. At the gelation point, this distribution decays algebraically, U_k ~ 1/(4k) for k>>1. As a result, the total number of unicyclic components grows logarithmically with the system size.Comment: 4 pages, 2 figure

    On the probability of planarity of a random graph near the critical point

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    Consider the uniform random graph G(n,M)G(n,M) with nn vertices and MM edges. Erd\H{o}s and R\'enyi (1960) conjectured that the limit \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} exists and is a constant strictly between 0 and 1. \L uczak, Pittel and Wierman (1994) proved this conjecture and Janson, \L uczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact probability of a random graph being planar near the critical point M=n/2M=n/2. For each λ\lambda, we find an exact analytic expression for p(λ)=limnPrG(n,n2(1+λn1/3))isplanar. p(\lambda) = \lim_{n \to \infty} \Pr{G(n,\textstyle{n\over 2}(1+\lambda n^{-1/3})) is planar}. In particular, we obtain p(0)0.99780p(0) \approx 0.99780. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of G(n,n2)G(n,\textstyle{n\over 2}) being series-parallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.Comment: 10 pages, 1 figur

    Creation and Growth of Components in a Random Hypergraph Process

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    Denote by an \ell-component a connected bb-uniform hypergraph with kk edges and k(b1)k(b-1) - \ell vertices. We prove that the expected number of creations of \ell-component during a random hypergraph process tends to 1 as \ell and bb tend to \infty with the total number of vertices nn such that =o(nb3)\ell = o(\sqrt[3]{\frac{n}{b}}). Under the same conditions, we also show that the expected number of vertices that ever belong to an \ell-component is approximately 121/3(b1)1/31/3n2/312^{1/3} (b-1)^{1/3} \ell^{1/3} n^{2/3}. As an immediate consequence, it follows that with high probability the largest \ell-component during the process is of size O((b1)1/31/3n2/3)O((b-1)^{1/3} \ell^{1/3} n^{2/3}). Our results give insight about the size of giant components inside the phase transition of random hypergraphs.Comment: R\'{e}sum\'{e} \'{e}tend

    On the length of a random minimum spanning tree

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    We study the expected value of the length LnL_n of the minimum spanning tree of the complete graph KnK_n when each edge ee is given an independent uniform [0,1][0,1] edge weight. We sharpen the result of Frieze \cite{F1} that \lim_{n\to\infty}\E(L_n)=\z(3) and show that \E(L_n)=\z(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}} where c1,c2c_1,c_2 are explicitly defined constants.Comment: Added next term and two co-author

    Forbidden Subgraphs in Connected Graphs

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    Given a set ξ={H1,H2,...}\xi=\{H_1,H_2,...\} of connected non acyclic graphs, a ξ\xi-free graph is one which does not contain any member of % \xi as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let {\gr{W}}_{k,\xi} be theexponential generating function (EGF for brief) of connected ξ\xi-free graphs of excess equal to kk (k1k \geq 1). For each fixed ξ\xi, a fundamental differential recurrence satisfied by the EGFs {\gr{W}}_{k,\xi} is derived. We give methods on how to solve this nonlinear recurrence for the first few values of kk by means of graph surgery. We also show that for any finite collection ξ\xi of non-acyclic graphs, the EGFs {\gr{W}}_{k,\xi} are always rational functions of the generating function, TT, of Cayley's rooted (non-planar) labelled trees. From this, we prove that almost all connected graphs with nn nodes and n+kn+k edges are ξ\xi-free, whenever k=o(n1/3)k=o(n^{1/3}) and ξ<|\xi| < \infty by means of Wright's inequalities and saddle point method. Limiting distributions are derived for sparse connected ξ\xi-free components that are present when a random graph on nn nodes has approximately n2\frac{n}{2} edges. In particular, the probability distribution that it consists of trees, unicyclic components, ......, (q+1)(q+1)-cyclic components all ξ\xi-free is derived. Similar results are also obtained for multigraphs, which are graphs where self-loops and multiple-edges are allowed

    On-line list colouring of random graphs

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    In this paper, the on-line list colouring of binomial random graphs G(n,p) is studied. We show that the on-line choice number of G(n,p) is asymptotically almost surely asymptotic to the chromatic number of G(n,p), provided that the average degree d=p(n-1) tends to infinity faster than (log log n)^1/3(log n)^2n^(2/3). For sparser graphs, we are slightly less successful; we show that if d>(log n)^(2+epsilon) for some epsilon>0, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of C, where C in [2,4], depending on the range of d. Also, for d=O(1), the on-line choice number is by at most a multiplicative constant factor larger than the chromatic number
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