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

The difference between the domination number and the minus domination number of a cubic graph

AbstractThe closed neighborhood of a vertex subset S of a graph G = (V, E), denoted as N[S], is defined as the union of S and the set of all the vertices adjacent to some vertex of S. A dominating set of a graph G = (V, E) is defined as a set S of vertices such that N[S] = V. The domination number of a graph G, denoted as γ(G), is the minimum possible size of a dominating set of G. A minus dominating function on a graph G = (V, E) is a function g : V → {−1, 0, 1} such that g(N[v]) ≥ 1 for all vertices. The weight of a minus dominating function g is defined as g(V) =ΣvϵVg(v). The minus domination number of a graph G, denoted as γ−(G), is the minimum possible weight of a minus dominating function on G. It is well known that γ−(G) ≤ γ(G). This paper is focused on the difference between γ(G) and γ−(G) for cubic graphs. We first present a graph-theoretic description of γ−(G). Based on this, we give a necessary and sufficient condition for γ(G) −γ−(G) ≥ k. Further, we present an infinite family of cubic graphs of order 18k + 16 and with γ(G) −γ−(G) ≥

Lower bounds on the signed (total) kk-domination number depending on the clique number

Let $G$ be a graph with vertex set $V(G)$‎. ‎For any integer $k\ge 1$‎, ‎a signed (total) $k$-dominating function‎ ‎is a function $f‎: ‎V(G) \rightarrow‎ \{ -1, ‎1\}$ satisfying $\sum_{x\in N[v]}f(x)\ge k$ ($\sum_{x\in N(v)}f(x)\ge k$)‎ ‎for every $v\in V(G)$‎, ‎where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)\cup\{v\}$‎. ‎The minimum of the values‎ ‎$\sum_{v\in V(G)}f(v)$‎, ‎taken over all signed (total) $k$-dominating functions $f$‎, ‎is called the signed (total)‎ ‎$k$-domination number‎. ‎The clique number of a graph $G$ is the maximum cardinality of a complete subgraph of $G$‎. ‎In this note we present some new sharp lower bounds on the signed (total) $k$-domination number‎ ‎depending on the clique number of the graph‎. ‎Our results improve some known bounds

Some remarks on domination in cubic graphs

We study three recently introduced numerical invariants of graphs, namely, the signed domination number γs, the minus domination number γ- and the majority domination number γmaj. An upper bound for γs and lower bounds for γ- and γmaj are found, in terms of the order of the graph