173 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    The hitting time of clique factors

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    In a recent paper, Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let r3r \ge 3 and let nn be divisible by rr. Then, in the random rr-uniform hypergraph process on nn vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we transfer this hitting time result to the setting of clique factors in the random graph process: At the time that the last vertex joins a copy of the complete graph KrK_r, the random graph process contains a KrK_r-factor. Our proof draws on a novel sequence of couplings, extending techniques of Riordan and the first author. An analogous result is proved for clique factors in the ss-uniform hypergraph process (s3s \ge 3)

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Paths and cycles in graphs and hypergraphs

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    In this thesis we present new results in graph and hypergraph theory all of which feature paths or cycles. A kk-uniform tight cycle Cn(k)C^{(k)}_n is a kk-uniform hypergraph on nn vertices with a cyclic ordering of its vertices such that the edges are all kk-sets of consecutive vertices in the ordering. We consider a generalisation of Lehel's Conjecture, which states that every 2-edge-coloured complete graph can be partitioned into two cycles of distinct colour, to kk-uniform hypergraphs and prove results in the 4- and 5-uniform case. For a kk-uniform hypergraph~HH, the Ramsey number r(H){r(H)} is the smallest integer NN such that any 2-edge-colouring of the complete kk-uniform hypergraph on NN vertices contains a monochromatic copy of HH. We determine the Ramsey number for 4-uniform tight cycles asymptotically in the case where the length of the cycle is divisible by 4, by showing that r(Cn(4))r(C^{(4)}_n) = (5+oo(1))nn. We prove a resilience result for tight Hamiltonicity in random hypergraphs. More precisely, we show that for any γ\gamma >0 and kk \geq 3 asymptotically almost surely, every subgraph of the binomial random kk-uniform hypergraph G(k)(n,nγ1)G^{(k)}(n, n^{\gamma -1}) in which all (k1)(k-1)-sets are contained in at least (12+2γ)pn(\frac{1}{2}+2\gamma)pn edges has a tight Hamilton cycle. A random graph model on a host graph HH is said to be 1-independent if for every pair of vertex-disjoint subsets A,BA,B of E(H)E(H), the state of edges (absent or present) in AA is independent of the state of edges in BB. We show that pp = 4 - 23\sqrt{3} is the critical probability such that every 1-independent graph model on Z2×Kn\mathbb{Z}^2 \times K_n where each edge is present with probability at least pp contains an infinite path

    Large Y3,2 Y_{3,2} -tilings in 3-uniform hypergraphs

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    Let Y3,2Y_{3,2} be the 3-graph with two edges intersecting in two vertices. We prove that every 3-graph H H on n n vertices with at least max{(4αn3),(n3)(nαn3)}+o(n3) \max \left \{ \binom{4\alpha n}{3}, \binom{n}{3}-\binom{n-\alpha n}{3} \right \}+o(n^3) edges contains a Y3,2Y_{3,2}-tiling covering more than 4αn 4\alpha n vertices, for sufficiently large n n and 0<α<1/40<\alpha< 1/4. The bound on the number of edges is asymptotically best possible and solves a conjecture of the authors for 3-graphs that generalizes the Matching Conjecture of Erd\H{o}s

    MCMC Sampling of Directed Flag Complexes with Fixed Undirected Graphs

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    Constructing null models to test the significance of extracted information is a crucial step in data analysis. In this work, we provide a uniformly sampleable null model of directed graphs with the same (or similar) number of simplices in the flag complex, with the restriction of retaining the underlying undirected graph. We describe an MCMC-based algorithm to sample from this null model and statistically investigate the mixing behaviour. This is paired with a high-performance, Rust-based, publicly available implementation. The motivation comes from topological data analysis of connectomes in neuroscience. In particular, we answer the fundamental question: are the high Betti numbers observed in the investigated graphs evidence of an interesting topology, or are they merely a byproduct of the high numbers of simplices? Indeed, by applying our new tool on the connectome of C. Elegans and parts of the statistical reconstructions of the Blue Brain Project, we find that the Betti numbers observed are considerable statistical outliers with respect to this new null model. We thus, for the first time, statistically confirm that topological data analysis in microscale connectome research is extracting statistically meaningful information

    Transversals via regularity

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    Given graphs G1,,GsG_1,\ldots,G_s all on the same vertex set and a graph HH with e(H)se(H) \leq s, a copy of HH is transversal or rainbow if it contains at most one edge from each GcG_c. When s=e(H)s=e(H), such a copy contains exactly one edge from each GiG_i. We study the case when HH is spanning and explore how the regularity blow-up method, that has been so successful in the uncoloured setting, can be used to find transversals. We provide the analogues of the tools required to apply this method in the transversal setting. Our main result is a blow-up lemma for transversals that applies to separable bounded degree graphs HH. Our proofs use weak regularity in the 33-uniform hypergraph whose edges are those xycxyc where xyxy is an edge in the graph GcG_c. We apply our lemma to give a large class of spanning 33-uniform linear hypergraphs HH such that any sufficiently large uniformly dense nn-vertex 33-uniform hypergraph with minimum vertex degree Ω(n2)\Omega(n^2) contains HH as a subhypergraph. This extends work of Lenz, Mubayi and Mycroft

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Resilience for Loose Hamilton Cycles

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    We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum dd-degree threshold for loose Hamiltonicity relative to the random kk-uniform hypergraph Hk(n,p)H_k(n,p) coincides with its dense analogue whenever pn(k1)/2+o(1)p \geq n^{- (k-1)/2+o(1)}. The value of pp is approximately tight for d>(k+1)/2d>(k+1)/2. This is particularly interesting because the dense threshold itself is not known beyond the cases when dk2d \geq k-2.Comment: 33 pages, 3 figure
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