Let G and H be two graphs with the same vertex set V . It is well known that a graph G can be transformed into a graph H by a sequence of 2-switches if and only if every vertex of V has the same degree in both G and H . We study the problem of finding the minimum number of 2-switches for transforming G into H .

Answering a question of Alon, Ding, Oporowski and Vertigan [4], we show that there exists an absolute constant C such that every graph G with maximum degree 5 has a vertex partition into 2 parts, such that the subgraph induced by each part has no component of size greater than C. We obtain similar results for partitioning graphs of given maximum degree into k parts (k > 2) as well. A related th...

We consider cycles and paths in multigraphic realizations of a degree sequence Q. in particular we show that there exists a realization of d in which no cycle has order greater than three and no path has length greater than four. In addition we show which orders of cycles and which lengths of paths exist in some realization of d.

For a subset A of the set of positive integers, a graph G is called A-coverable if G has a cycle (a subgraph in which all vertices have even degree) which intersects all edge-cuts T in G with |T | ∈ A, and A is said to be coverable if all graphs are A-coverable. As a possible approach to the Dominating cycle conjecture, Kaiser and Škrekovski conjectured in [Cycles intersecting edge-cuts of pres...

The Narumi–Katayama index of a graph G is equal to the product of the degrees of the vertices of G. In this paper we consider a new version of the Narumi– Katayama index in which each vertex degree d is multiplied d times. We characterize the graphs extremal w.r.t. this new topological index.

Sacks [14] showed that every computably enumerable (c.e.) degree > 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov [1] showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan [9] proved the existence of a c.e. a > 0 which has no c.e. sp...

The degree of an edge e in a hypergraph G is the number of other edges of G intersecting e, and the maximum edge degree of G is the maximum over the degrees of its edges. A natural question is: Which bound on the maximum edge degree of an r-uniform hypergraph G provides that G is k-colorable? The classical result in this direction belongs to Erdős and Lovász. In their seminal paper [2] (where t...

A random geometric digraph Gn is constructed by taking {X1, X2, · · ·Xn} in R independently at random with a common bounded density function. Each vertex Xi is assigned at random a sector Si of central angle α with inclination Yi, in a circle of radius r (with vertex Xi as the origin). An arc is present from vertex Xi to Xj , if Xj falls in Si. Suppose k is fixed and {kn} is a sequence with 1 ≪...

For a general random intersection graph we show an approximation of the vertex degree distribution by a Poisson mixture. key words: intersection graph, random graph, degree distribution