Nonseparating vertices in tournaments with large minimum degree

On the number of nonseparating vertices in strongly connected local tournaments

A 9k Kernel for Nonseparating Independent Set in Planar Graphs

We study kernelization (a kind of efficient preprocessing) for NP-hard problems on planar graphs. Our main result is a kernel of size at most 9k vertices for the Planar Maximum Nonseparating Independent Set problem. A direct consequence of this result is that Planar Connected Vertex Cover has no kernel with at most (9/8 − ǫ)k vertices, for any ǫ > 0, assuming P 6= NP. We also show a very simple...

On the cycle structure of in-tournaments

An in-tournament is an oriented graph such that the in-neighborhood of every vertex induces a tournament. Therefore, in-tournaments are a generalization of local tournaments where, for every vertex, the set of inneighbors as well as the set of out-neighbors induce a tournament. While local tournaments have been intensively studied very little is known about in-tournaments. It is the purpose of ...

Disjoint 3 - cycles in tournaments : a proof of the 1 Bermond - Thomassen conjecture for tournaments ∗

5 We prove that every tournament with minimum out-degree at least 2k− 1 contains k disjoint 6 3-cycles. This provides additional support for the conjecture by Bermond and Thomassen that 7 every digraph D of minimum out-degree 2k − 1 contains k vertex disjoint cycles. We also prove 8 that for every > 0, when k is large enough, every tournament with minimum out-degree at least 9 (1.5+ )k contains...

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...