A graph G is said to have bandwidth at most b, if there exists a labeling of the vertices by 1, 2, . . . , n, so that |i − j| ≤ b whenever {i, j} is an edge of G. Recently, Böttcher, Schacht, and Taraz verified a conjecture of Bollobás and Komlós which says that for every positive r,∆, γ, there exists β such that if H is an n-vertex r-chromatic graph with maximum degree at most ∆ which has band...

Let Fk denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollobás conjectured that for every 2 > 0 and positive integer k there is an n(k, 2) such that every 2-edge-coloring of the complete graph of order n ≥ n(k, 2) which has at least 2n2 ) edges in each color contains a member of Fk. This ...

An n-by-n bipartite graph is H-saturated if the addition of any missing edge between its two parts creates a new copy of H. In 1964, Erdős, Hajnal and Moon made a conjecture on the minimum number of edges in a Ks,s-saturated bipartite graph. This conjecture was proved independently by Wessel and Bollobás in a more general, but ordered, setting: they showed that the minimum number of edges in a ...

We consider the following question of Bollobás: given an r-colouring of E(Kn), how large a k-connected subgraph can we find using at most s colours? We provide a partial solution to this problem when s = 1 (and n is not too small), showing that when r = 2 the answer is n−2k+2, when r = 3 the answer is ⌊n−k 2 ⌋+1 or ⌈n−k 2 ⌉ + 1, and when r − 1 is a prime power then the answer lies between n r−1...

In this paper we study the following problem of Bollobás and Scott: What is the smallest f(k,m) such that for any integer k ≥ 2 and any graph G with m edges, there is a partition V (G) = ⋃k i=1 Vi such that for 1 ≤ i 6= j ≤ k, e(Vi ∪Vj) ≤ f(k,m)? We show that f(k,m) < 1.6m/k + o(m), and f(k,m) < 1.5m/k + o(m) for k ≥ 23. (While the graph K1,n shows that f(k,m) ≥ m/(k − 1), which is 1.5m/k when ...

The “Fibonacci Dichotomy” of Kaiser and Klazar [15] was one of the first general results on the enumeration of permutation classes. It states that if there are fewer permutations of length n in a permutation class than the nth Fibonacci number, for any n, then the enumeration of the class is given by a polynomial for sufficiently large n. Since the Fibonacci Dichotomy was established for permut...

Jones polynomials and Kauuman polynomials are the most prominent invariants of knot theory. For alternating links, they are easily computable from the Tutte polynomials by a result of Thistlethwaite (1988), but in general one needs colored Tutte polynomials, as introduced by Bollobas and Riordan (1999). Knots and links can be presented as labeled planar graphs. The tree width of a link L is dee...

We translate the concept of succession rule and the ECO method into matrix notation, introducing the concept of a production matrix. Among other things, this allows us to combine our method with other enumeration techniques using matrices, such as the method of Riordan matrices. Moreover, we show that certain operations on production matrices correspond to well known operations on the numerical...

Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain q-analogues of Laguerre and Charlier polynomials. The moments of these orthogonal polynomials have combinatorial models in terms of crossings in permutations and set partitions. The aim of this article is to prove simple formulas for the moments of the q-Laguerre and the q-Charlier polynomials, in the style of the Touchard...