A ~ P3-decomposition of a directed graph D is a partition of the arcs of D into directed paths of length 2. We characterize symmetric digraphs that do not admit a ~ P3-decomposition. We show that the only 2-regular, connected directed graphs that do not admit a ~ P3-decomposition are obtained from undirected odd cycles by replacing each edge by two oppositely directed arcs. In both cases, we gi...

We show that the edges of any graph $G$ containing two edge-disjoint spanning trees can be blue/red coloured so blue and red graphs are connected degrees at each vertex differ by most four. This improves a result H\"orsch. discuss variations question for digraphs, infinite computational question, resolve further questions H\"orsch in negative.

We discuss relations between several known (some false, some open) conjectures on 3-edgeconnected, cubic graphs and how they all fit into the same framework related to cuts in matchings. We then provide a construction of 3-edge-connected digraphs satisfying the property that for every even subgraph E, the graph obtained by contracting the edges of E is not strongly connected. This disproves a r...

Let D be a finite digraph, and let V0, . . . , Vk−1 be nonempty subsets of V (D). The (strong form of) Edmonds’ branching theorem states that there are pairwise edge-disjoint spanning branchings B0, . . . ,Bk−1 in D such that the root set of Bi is Vi (i = 0, . . . , k − 1) if and only if for all ∅ 6= X ⊆ V (D) the number of ingoing edges of X is greater than or equal to the number of sets Vi di...

The main subject of the thesis is connectivity augmentation: we would like to make a given graphk-connected by adding a minimum number of new edges. There are four basic problems in thisfield, since one might consider both edgeand node-connectivity augmentation in both graphsand digraphs. The thesis wishes to contribute to three out of these four problems: directed-and undirecte...

Let G=(V,E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection π of edge-disjoint subgraphs G_1,G_2,…,G_n of G such that every edge of G belongs to exactly one G_i,(1≤i ≤n). The decomposition 〖π={G〗_1,G_2,…,G_n} of a connected graph G is said to be a distinct edge geodetic decomposition if g_1 (G_i )≠g_1 (G_j ),(1≤i≠j≤n). The maximum cardinality of π...