7 n 2  ≤ WW (G1) + WW (G2) + WW (G3) ≤ 2  n + 2 4  + n 2  + 4(n − 1). The corresponding extremal graphs are characterized. Published by Elsevier B.V.

We solve several conjectures and open problems from a recent paper by Acharya [2]. Some of our results are relatives of the Nordhaus–Gaddum theorem, concerning the sum of domination parameters in hypergraphs and their complements. (A dominating set in H is a vertex set D ⊆ X such that, for every vertex x ∈ X\D there exists an edge E ∈ E with x ∈ E and E∩D 6= ∅.) As an example, it is shown that ...

‎The Steiner distance of a graph‎, ‎introduced by Chartrand‎, ‎Oellermann‎, ‎Tian and Zou in 1989‎, ‎is a natural generalization of the‎ ‎concept of classical graph distance‎. ‎For a connected graph $G$ of‎ ‎order at least $2$ and $Ssubseteq V(G)$‎, ‎the Steiner‎ ‎distance $d(S)$ among the vertices of $S$ is the minimum size among‎ ‎all connected subgraphs whose vertex sets contain $S$‎. ‎Let $...

The topic of this paper is the hat problem in which each of n players is uniformly and independently fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to...

Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...

A graph G is said to have property Pm if it contains no subdivision of Km+1 and no subdivision of Kdm=2e+1;bm=2c+1. Chartrand et al. (J. Combin Theory 10 (1971) 12–41) (see also Problem 6.3 in Jensen and Toft (Graph Coloring Problems, Wiley, New York, 1995) conjectured that the set of vertices (respectively, edges) of any graph with property Pm can be partitioned into m−n+1 subsets such that ea...

(Received September 1, 2009) Abstract. For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v ∈ V (G), the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) 6= NC(v) for every pair u, v of adjacent vertices of G. The minimum number of c...

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) = ∅ 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 {f1, f2, ....

The generalized k-connectivity κk(G) of a graph G was introduced by Chartrand et al. in 1984. It is natural to introduce the concept of generalized k-edge-connectivity, λk(G). For general k, the generalized k-edgeconnectivity of a complete graph is obtained. For k ≥ 3, tight upper and lower bounds of κk(G) and λk(G) are given for a connected graph G of order n, namely, 1 ≤ κk(G) ≤ n− k2 and 1 ≤...

Given a graph G, by a Grundy k-coloring of G we mean any proper k-vertex coloring of G such that for each two colors i and j, i < j , every vertex ofG colored by j has a neighbor with color i. The maximum k for which there exists a Grundy k-coloring is denoted by (G) and called Grundy (chromatic) number of G. We first discuss the fixed-parameter complexity of determining (G) k, for any fixed in...