Highly linked tournaments with large minimum out-degree

Nonseparating vertices in tournaments with large minimum degree

Searching for Maximum Out-Degree Vertices in Tournaments

A vertex x in a tournament T is called a king if for every vertex y of T there is a directed path from x to y of length at most 2. It is not hard to show that every vertex of maximum out-degree in a tournament is a king. However, tournaments may have kings which are not vertices of maximum out-degree. A binary inquiry asks for the orientation of the edge between a pair of vertices and receives ...

Minimum degree conditions for H-linked graphs

For a fixed multigraph H with vertices w1, . . . , wm, a graph G is H-linked if for every choice of vertices v1, . . . , vm in G, there exists a subdivision of H in G such that vi is the branch vertex representing wi (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs. Given a connected multigraph H with k edges and minimum degree at least two and n 7.5k, we...

Minimum degree and pan-k-linked graphs

For a k-linked graph G and a vector E S of 2k distinct vertices of G, an E S-linkage is a set of k vertex-disjoint paths joining particular vertices of E S. Let T denote theminimum order of an E S-linkage in G. A graph G is said to be pan-k-linked if it is k-linked and for all vectors E S of 2k distinct vertices of G, there exists an E S-linkage of order t for all t such that T ≤ t ≤ |V (G)|. W...

Two proofs of the Bermond-Thomassen conjecture for tournaments with bounded minimum in-degree

The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r−1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r = 2, and very recently the conjecture was proved for the case where r = 3. It is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we pre...