Maximum Number Of Vertex-disjoint Cycles Research Articles

On extremal factors of de Bruijn-like graphs

In 1972, Mykkeltveit confirmed Golomb’s conjecture on the de Bruijn graphs: he proved that the pure cycling register rule yields the maximum number of vertex-disjoint cycles in de Bruijn graphs. We show that this result encompasses the tensor product of the de Bruijn graph for strings of length n with a simple cycle of size k, when n divides k or vice versa. Furthermore, we give counting formulas for an array of cycling register rules, which includes Golomb’s well-studied linear register rules.

A note on disjoint cycles

We prove tight polynomial-time computable lower and upper bounds on the maximum number of vertex-disjoint cycles of a given graph in terms of its cycle rank and maximum genus.

Cycles in a tournament with pairwise zero, one or two given vertices in common

Chen et al. [Partitioning vertices of a tournament into independent cycles, J. Combin. Theory Ser. B 83 (2001) 213–220] proved that every k -connected tournament with at least 8 k vertices admits k vertex-disjoint cycles spanning the vertex set, which answered a question posed by Bollobas. In this paper, we prove, as a consequence of a more general result, that every k -connected tournament of diameter at least 4 contains k vertex-disjoint cycles spanning the vertex set. Then, for a connected tournament of diameter at most 3, we determine a relation between the maximum number of vertex-disjoint cycles and the maximum number of vertex-disjoint cycles spanning the vertex set of T. Also, by using a lemma of Chen et al. [Partitioning vertices of a tournament into independent cycles, J. Combin. Theory Ser. B 83 (2001) 213–220], we prove that a k -connected tournament of order at least 5 k - 3 , of diameter distinct from 3 (resp. 3) admits k (resp. k - 1 ) vertex-disjoint cycles spanning the vertex set of T, with only one exception. Finally, we give results on cycles with pairwise one or two vertices in common. A few open problems are raised.

Packing cycles in graphs, II

A graph G packs if for every induced subgraph H of G, the maximum number of vertex-disjoint cycles in H is equal to the minimum number of vertices whose deletion from H results in a forest. The purpose of this paper is to characterize all graphs that pack.

On extremal factors of the de Bruijn graph

An outstanding problem in combinatorial theory is to determine the maximum number of vertex-disjoint cycles in the n-th order de Bruijn graph G n . A related problem is to determine the minimum number of vertices which, if removed from G n , will leave a graph with no cycles. Both problems are discussed and it is shown that for n ≤ 8 the same number is attained for both. Further results indicate that the same is likely to be true for all values of n.