1,487 research outputs found

    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

    A new upper bound on the game chromatic index of graphs

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    We study the two-player game where Maker and Breaker alternately color the edges of a given graph GG with kk colors such that adjacent edges never get the same color. Maker's goal is to play such that at the end of the game, all edges are colored. Vice-versa, Breaker wins as soon as there is an uncolored edge where every color is blocked. The game chromatic index χg(G)\chi'_g(G) denotes the smallest kk for which Maker has a winning strategy. The trivial bounds Δ(G)χg(G)2Δ(G)1\Delta(G) \le \chi_g'(G) \le 2\Delta(G)-1 hold for every graph GG, where Δ(G)\Delta(G) is the maximum degree of GG. In 2008, Beveridge, Bohman, Frieze, and Pikhurko proved that for every δ>0\delta>0 there exists a constant c>0c>0 such that χg(G)(2c)Δ(G)\chi'_g(G) \le (2-c)\Delta(G) holds for any graph with Δ(G)(12+δ)v(G)\Delta(G) \ge (\frac{1}{2}+\delta)v(G), and conjectured that the same holds for every graph GG. In this paper, we show that χg(G)(2c)Δ(G)\chi'_g(G) \le (2-c)\Delta(G) is true for all graphs GG with Δ(G)Clogv(G)\Delta(G) \ge C \log v(G). In addition, we consider a biased version of the game where Breaker is allowed to color bb edges per turn and give bounds on the number of colors needed for Maker to win this biased game.Comment: 17 page

    New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix

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    The purpose of this article is to improve existing lower bounds on the chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency matrix sorted in non-increasing order. First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / - sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case m=1m=1. We provide several examples for which the new bound exceeds the {\sc Hoffman} lower bound. Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^- are the sums of the squares of positive and negative eigenvalues, respectively. To corroborate this conjecture, we prove the weaker bound chi >= S^+/S^-. We show that the conjectured lower bound is tight for several families of graphs. We also performed various searches for a counter-example, but none was found. Our proofs rely on a new technique of converting the adjacency matrix into the zero matrix by conjugating with unitary matrices and use majorization of spectra of self-adjoint matrices. We also show that the above bounds are actually lower bounds on the normalized orthogonal rank of a graph, which is always less than or equal to the chromatic number. The normalized orthogonal rank is the minimum dimension making it possible to assign vectors with entries of modulus one to the vertices such that two such vectors are orthogonal if the corresponding vertices are connected. All these bounds are also valid when we replace the adjacency matrix A by W * A where W is an arbitrary self-adjoint matrix and * denotes the Schur product, that is, entrywise product of W and A

    Logical limit laws for minor-closed classes of graphs

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    Let G\mathcal G be an addable, minor-closed class of graphs. We prove that the zero-one law holds in monadic second-order logic (MSO) for the random graph drawn uniformly at random from all {\em connected} graphs in G\mathcal G on nn vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in G\mathcal G on nn vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface SS. We also prove that the closure of the set of limiting probabilities is always the finite union of at least two disjoint intervals, and that it is the same for FO and MSO. For the classes of forests and planar graphs we are able to determine the closure of the set of limiting probabilities precisely. For planar graphs it consists of exactly 108 intervals, each of length 5106\approx 5\cdot 10^{-6}. Finally, we analyse examples of non-addable classes where the behaviour is quite different. For instance, the zero-one law does not hold for the random caterpillar on nn vertices, even in FO.Comment: minor changes; accepted for publication by JCT

    The biased odd cycle game

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    In this paper we consider biased Maker-Breaker games played on the edge set of a given graph GG. We prove that for every δ>0\delta>0 and large enough nn, there exists a constant kk for which if δ(G)δn\delta(G)\geq \delta n and χ(G)k\chi(G)\geq k, then Maker can build an odd cycle in the (1:b)(1:b) game for b=O(nlog2n)b=O(\frac{n}{\log^2 n}). We also consider the analogous game where Maker and Breaker claim vertices instead of edges. This is a special case of the following well known and notoriously difficult problem due to Duffus, {\L}uczak and R\"{o}dl: is it true that for any positive constants tt and bb, there exists an integer kk such that for every graph GG, if χ(G)k\chi(G)\geq k, then Maker can build a graph which is not tt-colorable, in the (1:b)(1:b) Maker-Breaker game played on the vertices of GG?Comment: 10 page
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