Trees through specified vertices

Trees through specified vertices

We prove a conjecture of Horak that can be thought of as an extension of classical results including Dirac’s theorem on the existence of Hamiltonian cycles. Namely, we prove for 1 ≤ k ≤ n − 2 if G is a connected graph with A ⊂ V (G) such that dG(v) ≥ k for all v ∈ A, then there exists a subtree T of G such that V (T ) ⊃ A and dT (v) ≤ ⌈ n−1 k ⌉ for all v ∈ A.

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.