Oriented Cycles in Digraphs of Large Outdegree

Subdivisions of oriented cycles in digraphs with large chromatic number

A note on complete subdivisions in digraphs of large outdegree

Mader conjectured that for all ` there is an integer δ(`) such that every digraph of minimum outdegree at least δ(`) contains a subdivision of a transitive tournament of order `. In this note we observe that if the minimum outdegree of a digraph is sufficiently large compared to its order then one can even guarantee a subdivision of a large complete digraph. More precisely, let ~ G be a digraph...

Orientations of hamiltonian cycles in large digraphs

We prove that, with some exceptions, every digraph with n 3 9 vertices and at least ( n 1) (n 2) + 2 arcs contains all orientations of a Hamiltonian cycle.

Short Cycles in Digraphs with Local Average Outdegree at Least Two

Suppose G is a strongly connected digraph with order n girth g and diameter d. We prove that d + g ≤ n if G contains no arcs (u, v) with deg(u) = 1 and deg(v) ≤ 2. Caccetta and Häggkvist showed in 1978 that any digraph of order n with minimum outdegree 2 contains a cycle of length at most dn/2e. Applying the abovementioned result, we improve their result by replacing the minimum outdegree condi...

0n removable cycles in graphs and digraphs

In this paper we define the removable cycle that, if $Im$ is a class of graphs, $Gin Im$, the cycle $C$ in $G$ is called removable if $G-E(C)in Im$. The removable cycles in Eulerian graphs have been studied. We characterize Eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for Eulerian graph to have removable cycles h...