We prove theorems on edge splittings and edge-connectivity augmentation in directed hypergraphs, extending earlier results of Mader and Frank, respectively, on directed graphs. MSC Classification: 05C40, 05C65, 05C85, 05C20

In this paper, we obtain a sufficient condition for the existence of parity factors in a regular graph in terms of edge-connectivity. Moreover, we also show that our condition is sharp.

The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G. We prove that if G is a λ-edge-connected graph of order n, then the bounds μ(G) ≤ 2n/15 + 9, if λ = 5, 6, μ(G) ≤ n/9 + 10, if λ = 7 and μ(G) ≤ n/(λ + 1) + 5, if λ ≥ 8 hold. Our bounds are shown to be best possible and our results solve a problem of Plesńık [12].

It is known that the edge set of a 2-edge-connected 3-regular graph can be decomposed into paths of length 3. W. Li asked whether the edge set of every 2-edge-connected graph can be decomposed into paths of length at least 3. The graphs C3, C4, C5, and K4 − e have no such decompositions. We construct an infinite sequence {Fi}∞i=0 of nondecomposable graphs. On the other hand, we prove that every...

Katerinis and Tsikopoulos (Graphs. Combin. 12 (1996) 327) give su6cient conditions for a regular bipartite graph to have a perfect matching excluding a set of edges. In this paper, we give a necessary and su6cient condition for a bipartite graph to have an f-factor containing a set of edges and excluding another set of edges and discuss some applications of this condition. In particular, we obt...

The transformation graph G−++ of G is the graph with vertex set V (G) ∪ E(G) in which the vertex x and y are joined by an edge if one of the following conditions holds: (i) x, y ∈ V (G), and x and y are not adjacent in G, (ii) x, y ∈ E(G), and x and y are adjacent in G, (iii) one of x and y is in V (G) and the other is in E(G), and they are incident in G. In this paper, it is shown that for two...

In this note we give a characterization of the complete bipartite graphs which have an even (odd) [a, b]-factor. For general graphs we prove that an a-edge connected graph G with n vertices and with δ(G) ≥ max{a + 1, an a+b + a − 2} has an even [a, b]-factor, where a and b are even and 2 ≤ a ≤ b. With regard to the edge-connectivity this result is slightly better than one of the similar results...

Let G be a connected graph of order n, minimum degree δ(G), and edge-connectivity κ (G). The graph G ismaximally edge-connected if κ (G) = δ(G) and super edge-connected if every minimum edge-cut consists of edges incident with a vertex of minimum degree. A list (d1, . . . , dn) is graphic if there is a graph with vertices v1, . . . , vn such that d(vi) = di for 1 ≤ i ≤ n. A graphic list D is su...

Ota and Tokuda [2] gave a minimum degree condition for a K1,n-free graph to have a 2factor. Though their condition is best-possible, their sharpness examples have edge-connectivity one. In this paper, we improve their minimum degree condition for K1,n-free graphs with large connectivity or large edge-connectivity. Our bound is sharp, and together with Ota and Tokuda’s theorem, we completely det...

Celebrity games, a new model of network creation games is introduced. The specific features of this model are that players have different celebrity weights and that a critical distance is taken into consideration. The aim of any player is to be close (at distance less than critical) to the others, mainly to those with high celebrity weights. The cost of each player depends on the cost of establ...