Note on Disjoint Cycles in Multipartite Tournaments

In 1981, Bermond and Thomassen conjectured that for any positive integer $k$, every digraph with minimum out-degree at least $2k-1$ admits $k$ vertex-disjoint directed cycles. In this short paper, we verify the Bermond-Thomassen conjecture for triangle-free multipartite tournaments and 3-partite tournaments. Furthermore, we characterize 3-partite tournaments with minimum out-degree at least $2k-1$ ($k\geq 2$) such that in each set of $k$ vertex-disjoint directed cycles, every cycle has the same length.Comment: 9 pages, 0 figur

Balanced directed cycle designs based on groups

AbstractA balanced directed cycle design with parameters (υ, k, λ) is a decomposition of the complete directed multigraph λK̄υ into edge disjoint directed cycles C̄k. The paper elaborates on the ‘difference method’ for constructing these designs from groups and obtains some new constructions and classification results for designs with large automorphism groups

On disjoint directed cycles with prescribed minimum lengths

In this paper, we show that the k-Linkage problem is polynomial-time solvable for digraphs with circumference at most 2. We also show that the directed cycles of length at least 3 have the Erdős-Pósa Property : for every n, there exists an integer t_n such that for every digraph D, either D contains n disjoint directed cycles of length at least 3, or there is a set T of t_n vertices that meets every directed cycle of length at least 3. From these two results, we deduce that if F is the disjoint union of directed cycles of length at most 3, then one can decide in polynomial time if a digraph contains a subdivision of F

Parameterized Directed kk-Chinese Postman Problem and kk Arc-Disjoint Cycles Problem on Euler Digraphs

In the Directed $k$-Chinese Postman Problem ($k$-DCPP), we are given a connected weighted digraph $G$ and asked to find $k$ non-empty closed directed walks covering all arcs of $G$ such that the total weight of the walks is minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128) asked for the parameterized complexity of $k$-DCPP when $k$ is the parameter. We prove that the $k$-DCPP is fixed-parameter tractable. We also consider a related problem of finding $k$ arc-disjoint directed cycles in an Euler digraph, parameterized by $k$. Slivkins (ESA 2003) showed that this problem is W[1]-hard for general digraphs. Generalizing another result by Slivkins, we prove that the problem is fixed-parameter tractable for Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler digraphs remains W[1]-hard even for Euler digraphs