A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number k is that they have bounded bandwidth. A weaker well-known conjecture is that their maximum degree is bounded in terms of k. In this note we disprove these conjectures for every k ≥ 171, by providing examples of k-crossing-critical graphs with arbitrarily large maximum degree. A graph is k-crossing-cr...

McDiarmid and Reed [‘On the maximum degree of a random planar graph’, Combin. Probab. Comput. 17 (2008) 591–601] showed that the maximum degree Δn of a random labeled planar graph with n vertices satisfies with high probability (w.h.p.) c1 logn < Δn < c2 logn, for suitable constants 0 < c1 < c2. In this paper, we determine the precise asymptotics, showing in particular that w.h.p. |Δn − c logn|...

MICHAEL DRMOTA, OMER GIMÉNEZ and MARC NOY Combinatorics, Probability and Computing / Volume 20 / Issue 04 / July 2011, pp 529 ­ 570 DOI: 10.1017/S0963548311000198, Published online: 31 May 2011 Link to this article: http://journals.cambridge.org/abstract_S0963548311000198 How to cite this article: MICHAEL DRMOTA, OMER GIMÉNEZ and MARC NOY (2011). The Maximum Degree of Series­ Parallel Graphs. C...

We show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal varieties under monomial maps with finite fibers. The maximum likelihood estimator corresponding to such a variety is Kapranov’s Horn uniformization. This extends Kapranov’s characterization of A-discriminantal hypersurfaces to varieties of arbitrary codimension.

A graph is said to be Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. Using combinatorial and matrix-theoretic techniques, we identify, up to isomorphism, the 21 connected Laplacian integral graphs of maximum degree 3 on at least 6 vertices.