An efficient sum of squares nonnegativity certificate for quaternary quartic

We show that for any non-negative 4-ary quartic form $f$ there exists a product of two non-negative quadrics $q$ and $q'$ so that $qq'f$ is a sum of squares (s.o.s.) of quartics. As a step towards deciding whether just one $q$ always suffices to make $qf$ a s.o.s, we show that there exist non-s.o.s. non-negative 3-ary sextics $ac-b^2$, with $a$, $b$, $c$ of degrees 2, 3, 4, respectively.Comment: LaTeX, 8 pages (significantly expanded w.r.t. version 1

Edge-dominating cycles, k-walks and Hamilton prisms in 2K22K_2-free graphs

We show that an edge-dominating cycle in a $2K_2$-free graph can be found in polynomial time; this implies that every 1/(k-1)-tough $2K_2$-free graph admits a k-walk, and it can be found in polynomial time. For this class of graphs, this proves a long-standing conjecture due to Jackson and Wormald (1990). Furthermore, we prove that for any \epsilon>0 every (1+\epsilon)-tough $2K_2$-free graph is prism-Hamiltonian and give an effective construction of a Hamiltonian cycle in the corresponding prism, along with few other similar results.Comment: LaTeX, 8 page

Implementing Brouwer's database of strongly regular graphs

Andries Brouwer maintains a public database of existence results for strongly regular graphs on $n\leq 1300$ vertices. We implemented most of the infinite families of graphs listed there in the open-source software Sagemath, as well as provided constructions of the "sporadic" cases, to obtain a graph for each set of parameters with known examples. Besides providing a convenient way to verify these existence results from the actual graphs, it also extends the database to higher values of $n$.Comment: 18 pages, LaTe

The Extensions of the Generalized Quadrangle of Order (3, 9)

AbstractIt is shown that there is only one extension of GQ(3, 9) namely the one admitting the sporadic simple groupMcLas a flag-transitive automorphism group. The proof depends on a computer calculation

The isometries of the cut, metric and hypermetric cones

We show that the symmetry groups of the cut cone Cut(n) and the metric cone Met(n) both consist of the isometries induced by the permutations on {1,...,n}; that is, Is(Cut(n))=Is(Met(n))=Sym(n) for n>4. For n=4 we have Is(Cut(4))=Is(Met(4))=Sym(3)xSym(4). This is then extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, Is(Hyp(n))=Sym(n) for n>4, where Hyp(n) denotes the hypermetric cone.Comment: 8 pages, LaTeX, 2 postscript figure

The inverse moment problem for convex polytopes: implementation aspects

We give a detailed technical report on the implementation of the algorithm presented in Gravin et al. (Discrete & Computational Geometry'12) for reconstructing an $N$-vertex convex polytope $P$ in $\mathbb{R}^d$ from the knowledge of $O(Nd)$ of its moments