Coloring Graphs with Forbidden Minors

Hadwiger's conjecture from 1943 states that for every integer $t\ge1$, every graph either can be $t$-colored or has a subgraph that can be contracted to the complete graph on $t+1$ vertices. As pointed out by Paul Seymour in his recent survey on Hadwiger's conjecture, proving that graphs with no $K_7$ minor are $6$-colorable is the first case of Hadwiger's conjecture that is still open. It is not known yet whether graphs with no $K_7$ minor are $7$-colorable. Using a Kempe-chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence number two, we first give a very short and computer-free proof of a recent result of Albar and Gon\c{c}alves and generalize it to the next step by showing that every graph with no $K_t$ minor is $(2t-6)$-colorable, where $t\in\{7,8,9\}$. We then prove that graphs with no $K_8^-$ minor are $9$-colorable and graphs with no $K_8^=$ minor are $8$-colorable. Finally we prove that if Mader's bound for the extremal function for $K_p$ minors is true, then every graph with no $K_p$ minor is $(2t-6)$-colorable for all $p\ge5$. This implies our first result. We believe that the Kempe-chain method we have developed in this paper is of independent interest

Renormalization: an advanced overview

We present several approaches to renormalization in QFT: the multi-scale analysis in perturbative renormalization, the functional methods \`a la Wetterich equation, and the loop-vertex expansion in non-perturbative renormalization. While each of these is quite well-established, they go beyond standard QFT textbook material, and may be little-known to specialists of each other approach. This review is aimed at bridging this gap.Comment: Review, 130 pages, 33 figures; v2: misprints corrected, refs. added, minor improvements; v3: some changes to sect. 5, refs. adde

Surface networks

© Copyright CASA, UCL. The desire to understand and exploit the structure of continuous surfaces is common to researchers in a range of disciplines. Few examples of the varied surfaces forming an integral part of modern subjects include terrain, population density, surface atmospheric pressure, physico-chemical surfaces, computer graphics, and metrological surfaces. The focus of the work here is a group of data structures called Surface Networks, which abstract 2-dimensional surfaces by storing only the most important (also called fundamental, critical or surface-specific) points and lines in the surfaces. Surface networks are intelligent and “natural ” data structures because they store a surface as a framework of “surface ” elements unlike the DEM or TIN data structures. This report presents an overview of the previous works and the ideas being developed by the authors of this report. The research on surface networks has fou