Dismantling sparse random graphs

We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph has no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n tending to infinity, then the number in question is essentially the same for all values of k such that k tends to infinity but k=o(n).Comment: 7 page

On Minimum Saturated Matrices

Motivated by the work of Anstee, Griggs, and Sali on forbidden submatrices and the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let F be a family of k-row matrices. A matrix M is called F-admissible if M contains no submatrix G\in F (as a row and column permutation of G). A matrix M without repeated columns is F-saturated if M is F-admissible but the addition of any column not present in M violates this property. In this paper we consider the function sat(n,F) which is the minimum number of columns of an F-saturated matrix with n rows. We establish the estimate sat(n,F)=O(n^{k-1}) for any family F of k-row matrices and also compute the sat-function for a few small forbidden matrices.Comment: 31 pages, included a C cod

Hypergraph containers

We develop a notion of containment for independent sets in hypergraphs. For every $r$-uniform hypergraph $G$, we find a relatively small collection $C$ of vertex subsets, such that every independent set of $G$ is contained within a member of $C$, and no member of $C$ is large; the collection, which is in various respects optimal, reveals an underlying structure to the independent sets. The containers offer a straightforward and unified approach to many combinatorial questions concerned (usually implicitly) with independence. With regard to colouring, it follows that simple $r$-uniform hypergraphs of average degree $d$ have list chromatic number at least $(1/(r-1)^2 + o(1)) \log_r d$. For $r = 2$ this improves a bound due to Alon and is tight. For $r \ge 3$, previous bounds were weak but the present inequality is close to optimal. In the context of extremal graph theory, it follows that, for each $\ell$-uniform hypergraph $H$ of order $k$, there is a collection $C$ of $\ell$-uniform hypergraphs of order $n$ each with $o(n^k)$ copies of $H$, such that every $H$-free $\ell$-uniform hypergraph of order $n$ is a subgraph of a hypergraph in $C$, and $\log |C| \le c n^{\ell-1/m(H)} \log n$ where $m(H)$ is a standard parameter (there is a similar statement for induced subgraphs). This yields simple proofs, for example, for the number of $H$-free hypergraphs, and for the sparsity theorems of Conlon-Gowers and Schacht. A slight variant yields a counting version of the K{\L}R conjecture. Likewise, for systems of linear equations the containers supply, for example, bounds on the number of solution-free sets, and the existence of solutions in sparse random subsets. Balogh, Morris and Samotij have independently obtained related results.The first author was supported by a grant from the EPSRC.This is the author accepted manuscript. The final version is available from Springer at http://dx.doi.org/10.1007/s00222-014-0562-

Wavelength routing in optical networks of diameter two

AbstractWe consider optical networks with routing by wavelength division multiplexing. We show that wavelength switching is unnecessary in routings where communication paths use at most two edges. We then exhibit routings in some explicit pseudo-random graphs, showing that they achieve optimal performance subject to constraints on the number of edges and the maximal degree. We also observe the relative inefficiency of planar networks

Bounding the size of square-free subgraphs of the hypercube

AbstractWe investigate the maximum size of a subset of the edges of the n-cube that does not contain a square, or 4-cycle. The size of such a subset is trivially at most 3/4 of the total number of edges, but the proportion was conjectured by Erdős to be asymptotically 1/2. Following a computer investigation of the 4-cube and the 5-cube, we improve the known upper bound from 0.62284… to 0.62256… in the limit

On Some Cycles in Wenger Graphs

Let p be a prime, q be a power of p, and let F q be the field of q elements. For any positive integer n, the Wenger graph W n (q) is defined as follows: it is a bipartite graph with the vertex partitions being two copies of the (n + 1)-dimensional vector space F n+1 , and two vertices p = (p(1), . . . , p(n + 1)), and l = [l(1), . . . , l(n + 1)] q being adjacent if p(i) + l(i) = p(1)l(1) i−1 , for all i = 2, 3, . . . , n + 1

Monochromatic triangles in three-coloured graphs

In 1959, Goodman determined the minimum number of monochromatic triangles in a complete graph whose edge set is two-coloured. Goodman also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, we determine the minimum number of monochromatic triangles and the colourings which achieve this minimum in a sufficiently large three-coloured complete graph.Comment: Some data needed to verify the proof can be found at http://www.math.cmu.edu/users/jcumming/ckpsty