Dirk Meierling

Nordhaus–Gaddum bounds on the k-rainbow domatic number of a graph

Abstract For a positive integer k , a k -rainbow dominating function of a graph G is a function f from the vertex set V ( G ) to the set of all subsets of the set { 1 , 2 , … , k } such that for any vertex v ∈ V ( G ) with f ( v ) = 0 the condition ⋃ u ∈ N ( v ) f ( u ) = { 1 , 2 , … , k } is fulfilled, where N ( v ) is the neighborhood of v . The 1 -rainbow domination is the same as the ordinary domination. A set { f 1 , f 2 , … , f d } of k -rainbow dominating functions on G with the property that ∑ i = 1 d | f i ( v ) | ≤ k for each v ∈ V ( G ) is called a k -rainbow dominating family (of functions) on G . The maximum number of functions in a k -rainbow dominating family on G is the k -rainbow domatic number of G , denoted by d r k ( G ) . Note that d r 1 ( G ) is the classical domatic number d ( G ) . If G is a graph of order n and G ¯ is the complement of G , then we prove in this note for k ≥ 2 the Nordhaus–Gaddum inequality d r k ( G ) + d r k ( G ¯ ) ≤ n + 2 k − 2 . This improves the Nordhaus–Gaddum bound given by Sheikholeslami and Volkmann recently.

Independence and k-domination in graphs

A subset S of vertices of a graph G is k-dominating if every vertex not in S has at least k neighbours in S. The k-domination number γ k (G) is the minimum cardinality of a k-dominating set of G, and α(G) denotes the cardinality of a maximum independent set of G. Brooks well-known bound for the chromatic number χ and the inequality α(G)≥n(G)/χ(G) for a graph G imply that α(G)≥n(G)/Δ(G) when G is non-regular and α(G)≥n(G)/(Δ(G)+1) otherwise. In this paper, we present a new proof of this property and derive some bounds on γ k (G). In particular, we show that, if G is connected with δ(G)≥k then γ k (G)≤(Δ(G)−1)α(G) with the exception of G being a cycle of odd length or the complete graph of order k+1. Finally, we characterize the connected non-regular graphs G satisfying equality in these bounds and present a conjecture for the regular case.

A remark on the (2,2)-domination number

A subset D of the vertex set of a graph G is a (k; p)-dominating set if every vertex v 2 V (G) n D is within distance k to at least p vertices in D. The parameter k;p(G) denotes the minimum cardinality of a (k; p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that k;p(G) p p+k n(G) for any graph G with k(G) k + p 1, where the latter means that every vertex is within distance k to at least k + p 1 vertices other than itself. In 2005, Fischermann and Volkmann conrmed this conjecture for all integers k and p for the case that p is a multiple of k. In this paper we show that 2;2(G) (n(G) + 1)=2 for all connected graphs G and characterize all connected graphs with 2;2 = (n+1)=2. This means that for k = p = 2 we characterize all connected graphs for which the conjecture is true without the precondition that 2 3.

A lower bound for the distance k-domination number of trees

A subset D of vertices of a graph G = (V, E) is a distance k-dominating set for G if the distance between every vertex of V − D and D is at most k. The minimum size of a distance k-dominating set of G is called the distance k-domination number γk(G) of G. In this paper we prove that (2k + 1)γk(T) ≥ ¦V¦ + 2k − kn1 for each tree T = (V, E) with n1 leafs, and we characterize the class of trees that satisfy the equality (2k + 1)γk(T) = ¦V¦ + 2k − kn1. Our results generalize those of Lemanska [4] for k = 1 and of Cyman, Lemanska and Raczek [1] for k = 2.

The Erdős-Pósa Property for Long Circuits

For an integer i�� at least 3, we prove that if G is a graph containing no two vertex-disjoint circuits of length at least i��, then there is a set X of at most 53i��+292 vertices that intersects all circuits of length at least i��. Our result improves the bound 2i��+3 due to Birmele, Bondy, and Reed The Erdi��s-Posa property for long circuits, Combinatorica 27 2007, 135-145 who conjecture that i�� vertices always suffice.

Complementary cycles in regular multipartite tournaments, where one cycle has length five

The vertex set of a digraph D is denoted by V(D). A c-partite tournament is an orientation of a complete c-partite graph. In 1999, Yea conjectured that each regular c-partite tournament D with and contains a pair of vertex disjoint directed cycles of lengths 4 and . An example will demonstrate that Yeos conjecture is not true in general for regular 4-partite tournaments with two vertices in each partite. However, in all other cases we shall confirm this conjecture in affirmative.

All 2-connected in-tournaments that are cycle complementary

An in-tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. A digraph D is cycle complementary if there exist two vertex-disjoint directed cycles spanning the vertex set of D. Let D be a 2-connected in-tournament of order at least 8. In this paper we show that D is not cycle complementary if and only if it is 2-regular and has odd order.

Maximally edge-connected hypergraphs

The edge-connectivity of a connected graph or hypergraph is the minimum number of edges whose removal renders the graph or hypergraph, respectively, disconnected. The edge-connectivity of a (hyper) graph cannot exceed its minimum degree. For graphs, several sufficient conditions for equality of edge-connectivity and minimum degree are known. For example Chartrand (1966) showed that for every graph of order n and minimum degree at least n - 1 2 its edge-connectivity equals its minimum degree. We show that this and some other well-known sufficient conditions generalise to hypergraphs.

Cycles in complementary prisms

The complementary prism GG of a graph G arises from the disjoint union of G and the complement G of G by adding a perfect matching joining corresponding pairs of vertices in G and G. Partially answering a question posed by Haynes etal. (2007) we provide an efficient characterization of the circumference of the complementary prism TT of a tree T and show that TT has cycles of all lengths between 3 and its circumference. Furthermore, we prove that for a given graph of bounded maximum degree it can be decided in polynomial time whether its complementary prism is Hamiltonian.

A general method in the theory of domination in graphs

In this article, we present a method to derive inequalities involving various domination parameters in graphs. As an application, we determine several lower bounds for these domination parameters in trees in terms of the order and the number of leaves. Finally, we characterize the classes of extremal trees.