The minimum number of edges of an undirected graph covering a symmetric , supermodular set-function is determined. As a special case, we derive an extension of a theorem of J. Bang-Jensen and B. Jackson on hypergraph connectivity augmentation. 0. INTRODUCTION T. Watanabe and A. Nakamura 1987] proved a min-max formula for the minimum number of new edges whose addition to a given undirected graph...

The Cayley Isomorphism property for combinatorial objects was introduced by L. Babai in 1977. Since then it has been intensively studied for binary relational structures: graphs, digraphs, colored graphs etc. In this paper we study this property for oriented Cayley maps. A Cayley map is a Cayley graph provided by a cyclic rotation of its connection set. If the underlying graph is connected, the...

We study the following natural variant of the budgeted maximum coverage problem: We are given a budget B and a hypergraph G = (V,E), where each vertex has a non-negative cost and a non-negative profit. The goal is to select a set of hyperedges T ⊆ E such that the total cost of the vertices covered by T is at most B and the total profit of all covered vertices is maximized. Besides being a natur...

A digraph Γ is called n-Cayley digraph over a group G, if there exists a semiregular subgroup RG of Aut(Γ) isomorphic to G with n orbits. In this paper, we represent the adjacency matrix of Γ as a diagonal block matrix in terms of irreducible representations of G and determine its characteristic polynomial. As corollaries of this result we find: the spectrum of semi-Cayley graphs over abelian g...

A substantial amount of research in graph theory continues to concentrate on the existence of hamiltonian cycles and perfect matchings. A classic theorem of Dirac states that a sufficient condition for an n-vertex graph to be hamiltonian, and thus, for n even, to have a perfect matching, is that the minimum degree is at least n/2. Moreover, there are obvious counterexamples showing that this is...

Let T be the Cayley graph of a finitely generated free group F . Given two vertices in T consider all the walks of a given length between these vertices that at a certain time must follow a number of predetermined steps. We give formulas for the number of such walks by expressing the problem in terms of equations in F and solving the corresponding equations.

For an integer k ≥ 2 and a k-uniform hypergraph H, let δk−1(H) be the largest integer d such that every (k−1)-element set of vertices of H belongs to at least d edges of H. Further, let t(k, n) be the smallest integer t such that every k-uniform hypergraph on n vertices and with δk−1(H) ≥ t contains a perfect matching. The parameter t(k, n) has been completely determined for all k and large n d...