We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 0 < γ < 1/2 we find a constant c = c(γ) such that the following holds. Let G = (V,E) be a (pseudo)random directed graph on n vertices and with at least a linear number of edges, and let G′ be a subgraph of G with (1/2 + γ)|E| edges. Then G′ contains a directe...

A seminal result of Reed et al. [15] in 1996 states that the Erdős-Pósa property holds for directed cycles, i.e. for every integer n there is an integer t such that every directed graph G has n pairwise vertex disjoint directed cycles or contains a set T ⊆ V (G) of at most t vertices such that G−T contains no directed cycle. In this paper, we consider the Erdős-Pósa property for directed cycles...

Proof: ⇐: Fix a strongly-connected orientation D. For any non-empty U ⊂ V , we may choose u ∈ U and v ∈ V \ U . Since D is strongly connected, there is a directed u-v path and a directed v-u path. Thus |δ D(U)| ≥ 1 and |δ − D(U)| ≥ 1, implying |δG(U)| ≥ 2. ⇒: Since G is 2-edge-connected, it has an ear decomposition. We proceed by induction on the number of ears. If G is a cycle then we may orie...

A Hamilton cycle in a digraph is a cycle passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this, is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so called Posá ‘rotationextension’ te...

We study k-colored kernels in m-colored digraphs. An m-colored digraph D has k-colored kernel if there exists a subset K of its vertices such that (i) from every vertex v / ∈ K there exists an at most k-colored directed path from v to a vertex of K and (ii) for every u, v ∈ K there does not exist an at most k-colored directed path between them. In this paper, we prove that for every integer k ≥...

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. A special case of L-cycle covers are k-cycle covers for k ∈ N, where the length of each cycle must be at least k. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges. We com...

The concept of a line digraph is generalized to that of a directed path graph. The directed path graph → Pk(D) of a digraph D is obtained by representing the directed paths on k vertices of D by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in D form a directed path on k + 1 vertices or form a directed cycle on k vertices in D. Several properties of → P3(...

purpose: the present study aims at comparing the impact of two teaching/learning approaches-self- directed/paced learning and lecture/demonstration-based instruction-on mastering psychomotor skills among nursing students. material & method: thirty nursing students were selected for the study. they filled out questionnaires indicating the demographic factor, and mean exam scores achieved in the ...

in this paper, the effects of input temperature and compression ratio on the net output work and efficiency of the air standard cycles, i.e. atkinson cycle, diesel cycle and otto cycle are analyzed. we assumed that the compression and power processes are adiabatic and reversible and any convective, conductive and radiative heat transfer to cylinder wall during the heat rejection process may be ...

Let G be a multidigraph without loops. Let l i be the upper bounds for arcs a i ∈ A(G) to be visited by any closed directed walk in G. We prove that there exists a sequence of finite integers {l i } for which every arc (and every parallel number of arcs) reversal in G decreases the number of closed directed walks if and only if every arc belongs to an elementary directed cycle in G.