We show that every set of n points in general position has a minimum pseudo-triangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudo-triangulation whose maximum face degree is four (i.e. each interior face of this pseudo-triangulation has at most four vertices). Both degree bounds are tight. Minimum pseudo-trian...

We prove that any permutation group of degree n ≥ 4 has at most 5(n−1)/3 conjugacy classes.

Let G be a (finite or infinite) graph with vertex set V(G) and edge set E(G). A partition P = {vi: i E Z} of V(G) is called a k-partition if [Vi1 s k for every i E I. Let the graph Gp be defined by (i) V(Gp) = {vi: i E Z}, (ii) l$l$ E E(G,)exy E (G) for some x E l$, y E l$ (i #j). Given an infinite graph G, let deg,(G) denote the least d for which there exist a natural number k and a k-partitio...

We show that there exists a matching with 4m 5k+3 edges in a graph of degree k and m edges.

We prove that, for each degree c r.e. in and above O'3', the class of degrees x < 0' with x'3) = c is definable without parameters in 3{< 0'), the degrees below 0'. Indeed the same definitions work below any r.e. degree r in place of 0'. Thus for each r.e. degree r, Th(^(< r)) uniquely determines

Let G be the set that contains precisely the graphs on n vertices with maximum degree 3 for which there exists a 4-cycle system of their complement in Kn. In this paper G is completely characterized. © 2007 Elsevier B.V. All rights reserved.

We show that the intersection dimension of graphs with respect to several hereditary graph classes can be bounded as a function of the maximum degree. As an interesting special case, we show that the circular dimension of a graph with maximum degree ∆ is at most O(∆ log∆ log log∆). We also obtain bounds in terms of treewidth.

This paper analyzes the production of Catalan schwa by children and adults in two areas of Barcelona differing in degree of presence of Spanish (which does not have schwa in its sound inventory). The schwa production in different kinds of words or in different positions in the word provides evidence for the status of this vowel.

We prove a number of Turán and Ramsey type stability results for cycles, in particular, the following one: Let n ≥ 4, 0 < β ≤ 1/2 − 1/2n, and the edges of K ⌊(2−β)n⌋ be 2-colored so that no mononchromatic C n exists. Then, for some q ∈ ((1 − β) n − 1, n) , we may drop a vertex v so that in K ⌊(2−β)n⌋ − v one of the colors induces K q,⌊(2−β)n⌋−q−1 , while the other one induces K q ∪ K ⌊(2−β)n⌋−q...

In this note, we give an easy and short proof for the theorem by Park and Kim stating that the hypercompetition numbers of hypergraphs with maximum degree at most two is at most two.