The classical Hadwiger conjecture dating back to 1940s states that any graph of chromatic number at least r has the clique order as a minor. Hadwiger's is an example well-studied class problems asking how large minor one can guarantee in with certain restrictions. One problem this type asks what largest size on n vertices independence most r. If true would imply existence . Results Kühn and Ost...

We speed up marginal inference by ignoring factors that do not significantly contribute to overall accuracy. In order to pick a suitable subset of factors to ignore, we propose three schemes: minimizing the number of model factors under a bound on the KL divergence between pruned and full models; minimizing the KL divergence under a bound on factor count; and minimizing the weighted sum of KL d...

We introduce two new notions for hypergraphs, hypertree-depth and minors in hypergraphs. We characterise hypergraphs of bounded hypertree-depth by the ultramonotone robber and marshals game, by vertex-hyperrankings and by centred hypercolourings. Furthermore, we show that minors in hypergraphs are ‘well-behaved’ with respect to hypertree-depth and other hypergraph invariants, such as generalise...

Article history: Received 19 May 2011 Available online xxxx

Consider two locally finite rooted trees as equivalent if each of them is a topological minor of the other, with an embedding preserving the tree-order. Answering a question of van der Holst, we prove that there are uncountably many equivalence classes.

It is proved that there are functions f (r) and N(r, s) such that for every positive integer r , s, each graph G with average degree d(G) = 2|E(G)|/|V (G)| ≥ f (r), and with at least N(r, s) vertices has a minor isomorphic to Kr,s or to the union of s disjoint copies of Kr . © 2005 Published by Elsevier Ltd

It is known that if G is a graph that can be drawn without edges crossing in a surface with Euler characteristic ǫ, and k and d are positive integers such that k > 3 and d is sufficiently large in terms of k and ǫ, then G is (k, d)∗-colorable; that is, the vertices of G can be colored with k colors so that each vertex has at most d neighbors with the same color as itself. In this paper, the kno...

Let G be a graph of order n. Let K− l be the graph obtained from Kl by removing one edge. In this paper, we propose the following conjecture: Let G be a graph of order n ≥ lk with δ(G) ≥ (n−k+1) l−3 l−2 +k−1. Then G has k vertex-disjoint K− l . This conjecture is motivated by Hajnal and Szemerédi’s [6] famous theorem. In this paper, we verify this conjecture for l = 4.

Path-addition is an operation that takes a graph and adds an internally vertex-disjoint path between two vertices together with a set of supplementary edges. Path-additions are just the opposite of taking minors. We show that some classes of graphs are closed under path-addition, including non-planar, right angle crossing, fan-crossing free, quasi-planar, (aligned) bar 1-visibility, and interva...