(2K + 1)-Connected Tournaments with Large Minimum Out-Degree are K-Linked

Nonseparating vertices in tournaments with large minimum degree

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...

Optimal Consecutive-k-out-of-(2k+1): G Cycle

We present a complete proof for the invariant optimal assignment for consecutive-k-outof-(2k+1):G cycle, which was proposed by Zuo and Kao in 1990 with an incomplete proof, pointed out recently by Jalali, Hawkes, Cui and Hwang.

The (2k-1)-connected multigraphs with at most k-1 disjoint cycles

In 1963, Corrádi and Hajnal proved that for all k≥1 and n≥3k, every (simple) graph G on n vertices with minimum degree δ(G)≥2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k− 1)-connected multigraphs do not contain k disjoint cycles? Recently, the authors characterized the s...

Disjoint 3-Cycles in Tournaments: A Proof of The Bermond-Thomassen Conjecture for Tournaments

We prove that every tournament with minimum out-degree at least 2k− 1 contains k disjoint 3-cycles. This provides additional support for the conjecture by Bermond and Thomassen that every digraph D of minimum out-degree 2k − 1 contains k vertex disjoint cycles. We also prove that for every > 0, when k is large enough, every tournament with minimum out-degree at least (1.5+ )k contains k disjoin...