1,252 research outputs found

    On the chromatic number of a random hypergraph

    Get PDF
    We consider the problem of kk-colouring a random rr-uniform hypergraph with nn vertices and cncn edges, where kk, rr, cc remain constant as nn tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case r=2r=2, must have one of two easily computable values as nn tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs.Comment: 45 pages, 2 figures, revised versio

    On the spectrum of hypergraphs

    Full text link
    Here we study the spectral properties of an underlying weighted graph of a non-uniform hypergraph by introducing different connectivity matrices, such as adjacency, Laplacian and normalized Laplacian matrices. We show that different structural properties of a hypergrpah, can be well studied using spectral properties of these matrices. Connectivity of a hypergraph is also investigated by the eigenvalues of these operators. Spectral radii of the same are bounded by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by the eigenvalues of its connectivity matrices. We characterize different properties of a regular hypergraph characterized by the spectrum. Strong (vertex) chromatic number of a hypergraph is bounded by the eigenvalues. Cheeger constant on a hypergraph is defined and we show that it can be bounded by the smallest nontrivial eigenvalues of Laplacian matrix and normalized Laplacian matrix, respectively, of a connected hypergraph. We also show an approach to study random walk on a (non-uniform) hypergraph that can be performed by analyzing the spectrum of transition probability operator which is defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two different ways. We show that if the Laplace operator, Δ\Delta, on a hypergraph satisfies a curvature-dimension type inequality CD(m,K)CD (\mathbf{m}, \mathbf{K}) with m>1\mathbf{m}>1 and K>0\mathbf{K}>0 then any non-zero eigenvalue of Δ- \Delta can be bounded below by mKm1 \frac{ \mathbf{m} \mathbf{K}}{ \mathbf{m} -1 } . Eigenvalues of a normalized Laplacian operator defined on a connected hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph

    H\"older-type inequalities and their applications to concentration and correlation bounds

    Get PDF
    Let Yv,vV,Y_v, v\in V, be [0,1][0,1]-valued random variables having a dependency graph G=(V,E)G=(V,E). We show that E[vVYv]vV{E[Yvχbb]}bχb, \mathbb{E}\left[\prod_{v\in V} Y_{v} \right] \leq \prod_{v\in V} \left\{ \mathbb{E}\left[Y_v^{\frac{\chi_b}{b}}\right] \right\}^{\frac{b}{\chi_b}}, where χb\chi_b is the bb-fold chromatic number of GG. This inequality may be seen as a dependency-graph analogue of a generalised H\"older inequality, due to Helmut Finner. Additionally, we provide applications of H\"older-type inequalities to concentration and correlation bounds for sums of weakly dependent random variables.Comment: 15 page

    On the Chromatic Thresholds of Hypergraphs

    Full text link
    Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least c(V(H)r1)c \binom{|V(H)|}{r-1} has bounded chromatic number. This parameter has a long history for graphs (r=2), and in this paper we begin its systematic study for hypergraphs. {\L}uczak and Thomass\'e recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Tur\'an number is achieved uniquely by the complete (r+1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of nondegenerate hypergraphs whose Tur\'an number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle. In order to prove upper bounds we introduce the concept of fiber bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fiber bundle dimension, a structural property of fiber bundles that is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemer\'edi for graphs and might be of independent interest. Many open problems remain.Comment: 37 pages, 4 figure

    On some building blocks of hypergraphs: units, twin-units, regular, co-regular, and symmetric sets

    Full text link
    Here, we introduce and investigate different building blocks, named units, twin units, regular sets, symmetric sets, and co-regular sets in a hypergraph. Our work shows that the presence of these building blocks leaves certain traces in the spectrum and the corresponding eigenspaces of the connectivity operators associated with the hypergraph. We also show that, conversely, some specific footprints in the spectrum and in the corresponding eigenvectors retrace the presence of some of these building blocks in the hypergraph. The hypergraph remains invariant under the permutations among the vertices in some building blocks. These vertices behave similarly, in random walks on the hypergraph and play an important role in hypergraph automorphisms. Identifying similar vertices in certain building blocks results in a smaller hypergraph that contains some spectral information of the original hypergraph. The number of specific building blocks provides an upper bound of the chromatic number of the hypergraph. A pseudo metric is introduced to measure distances between vertices in the hypergraph by using one of the building blocks. Here, we use the concept of general connectivity operators of a hypergraph for our spectral study.Comment: Title is changed and some new section adde
    corecore