On Connected Components with Many Edges

Equisingular Families of Plane Curves with Many Connected Components

We present examples of equisingular families of complex projective plane curves with plural connected components that are not distinguished by the fundamental group of the complement.

On contractible edges in 3-connected graphs

The existence of contractible edges is a very useful tool in graph theory. For 3-connected graphs with at least six vertices, Ota and Saito (1988) prove that the set of contractible edges cannot be covered by two vertices. Saito (1990) prove that if a three-element vertex set S covers all contractible edges of a 3-connected graph G, then S is a vertex-cut of G provided that G has at least eight...

Color critical hypergraphs with many edges

We show that for all k ≥ 3, r > l ≥ 2 there exists constant c = c(k, r, l) such that for large enough n there exists a k-colour-critical r-uniform hypergraph on less than n vertices, having more than cn edges, and having no l-set of vertices occuring in more than one edge.

Colouring edges with many colours in cycles

The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arbp(G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|, p + 1) colours. In the particular c...

Connected Components on Distributed Memory