On cycles through specified vertices

On cycles through specified vertices

For a set X of vertices of a graph fulfilling local connectedness conditions the existence of a cycle containing X is proved. AMS classification: 05C38, 05C45, 05C35

Long cycles through specified edges and vertices

Edge disjoint cycles through specified vertices

We give a sufficient condition for a simple graph G to have k pairwise edge-disjoint cycles, each of which contains a prescribed set W of vertices. The condition is that the induced subgraph G[W ] be 2k-connected, and that for any two vertices at distance two in G[W ], at least one of the two has degree at least |V (G)|/2 + 2(k − 1) in G. This is a common generalization of special cases previou...

Cycles through specified vertices in triangle-free graphs

Let G be a triangle-free graph with δ(G) ≥ 2 and σ4(G) ≥ |V (G)|+ 2. Let S ⊂ V (G) consist of less than σ4/4 + 1 vertices. We prove the following. If all vertices of S have degree at least three, then there exists a cycle C containing S. Both the upper bound on |S| and the lower bound on σ4 are best possible.

Heavy cycles passing through some specified vertices in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2-connected weighted graph such that the minimum weighted degree of G is at lea...