On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

A Sard theorem for Tame Set-Valued mappings

If $F$ is a set-valued mapping from $\R^n$ into $\R^m$ with closed graph, then $y\in \R^m$ is a critical value of $F$ if for some $x$ with $y\in F(x)$, $F$ is not metrically regular at $(x,y)$. We prove that the set of critical values of a set-valued mapping whose graph is a definable (tame) set in an $o$-minimal structure containing additions and multiplications is a set of dimension not greater than $m-1$ (resp. a porous set). As a corollary of this result we get that the collection of asymptotically critical values of a semialgebraic set-valued mapping has dimension not greater than $m-1$, thus extending to such mappings a corresponding result by Kurdyka-Orro-Simon for $C^1$ semialgebraic mappings. We also give an independent proof of the fact that a definable continuous real-valued function is constant on components of the set of its subdifferentiably critical points, thus extending to all definable functions a recent result of Bolte-Daniilidis-Lewis for globally subanalytic functions.Comment: 23

Topological minimal sets and their applications

In this article we introduce a definition of topological minimal sets, which is a generalization of that of Mumford-Shah-minimal sets. We prove some general properties as well as two existence theorems for topological minimal sets. As an application we prove the topological minimality of the union of two almost orthogonal planes in $\R^4$, and use it to improve the angle criterion under which the union of several higher dimensional planes is Almgren-minimal

Compactness and finite forcibility of graphons

Graphons are analytic objects associated with convergent sequences of graphs. Problems from extremal combinatorics and theoretical computer science led to a study of graphons determined by finitely many subgraph densities, which are referred to as finitely forcible. Following the intuition that such graphons should have finitary structure, Lovasz and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon is always compact. We disprove the conjecture by constructing a finitely forcible graphon such that the associated space is not compact. The construction method gives a general framework for constructing finitely forcible graphons with non-trivial properties