Discrepancy and Signed Domination in Graphs and Hypergraphs

For a graph G, a signed domination function of G is a two-colouring of the vertices of G with colours +1 and -1 such that the closed neighbourhood of every vertex contains more +1's than -1's. This concept is closely related to combinatorial discrepancy theory as shown by Fueredi and Mubayi [J. Combin. Theory, Ser. B 76 (1999) 223-239]. The signed domination number of G is the minimum of the sum of colours for all vertices, taken over all signed domination functions of G. In this paper, we present new upper and lower bounds for the signed domination number. These new bounds improve a number of known results.Comment: 12 page

Maximum Number of Minimum Dominating and Minimum Total Dominating Sets

Given a connected graph with domination (or total domination) number \gamma>=2, we ask for the maximum number m_\gamma and m_{\gamma,T} of dominating and total dominating sets of size \gamma. An exact answer is provided for \gamma=2and lower bounds are given for \gamma>=3.Comment: 6 page

The bondage number of graphs on topological surfaces and Teschner's conjecture

The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of the graph, and show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus. Also, we provide stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of the graph genera. This settles Teschner's Conjecture in positive for almost all graphs.Comment: 21 pages; Original version from January 201