Smoothed Analysis on Connected Graphs

Smoothed Analysis on Connected Graphs

The main paradigm of smoothed analysis on graphs suggests that for any large graph G in a certain class of graphs, perturbing slightly the edges of G at random (usually adding few random edges to G) typically results in a graph having much “nicer” properties. In this work we study smoothed analysis on trees or, equivalently, on connected graphs. Given an n-vertex connected graph G, form a rando...

On smoothed analysis in dense graphs and formulas

We study a model of random graphs, where a random instance is obtained by adding random edges to a large graph of a given density. The research on this model has been started by Bohman et al. in [3], [4]. Here we obtain a sharp threshold for the appearance of a fixed subgraph, and for certain Ramsey properties. We also consider a related model of random k-SAT formulas, where an instance is obta...

On 3*-connected graphs

Menger's Theorem states that in a 3-connected graph, any two vertices are joined by three openly disjoint paths. Here we consider 3-connected cubic graphs where two vertices exist so that the three disjoint paths between them contain all of the vertices of the graph (we call these graphs 3*connected); and also where the latter is true for ALL pairs of vertices (globally 3*-connected). A necessa...

Minicourse on Smoothed Analysis

On k-chromatically connected graphs

A graph G is chromatically k–connected if every vertex cutset induces a subgraph with chromatic number at least k. Thus, in particular each neighborhood has to induce a k–chromatic subgraph. In [3], Godsil, Nowakowski and Nešetřil asked whether there exists a k–chromatically connected graph such that every minimal cutset induces a subgraph with no triangles. We show that the answer is positive ...