Ernst J. Joubert

Nordhaus–Gaddum bounds for total domination

In the metadata of the chapter that will be visualized online, please replace the abstract with the following: “In this chapter, we consider Nordhaus- Gaddum inequalities involving the total domination number”

A lower bound for the packing chromatic number of the Cartesian product of cycles

Let G = (V, E) be a simple graph of order n and i be an integer with i ≥ 1. The set Xi ⊆ V(G) is called an i-packing if each two distinct vertices in Xi are more than i apart. A packing colouring of G is a partition X = {X1, X2, …, Xk} of V(G) such that each colour class Xi is an i-packing. The minimum order k of a packing colouring is called the packing chromatic number of G, denoted by χρ(G). In this paper we show, using a theoretical proof, that if q = 4t, for some integer t ≥ 3, then 9 ≤ χρ(C4 □ Cq). We will also show that if t is a multiple of four, then χρ(C4 □ Cq) = 9.

Restrained domination in cubic graphs

AbstractLet G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex in V−S is adjacent to a vertex in S and to a vertex in V−S. The restrained domination number of G, denoted γr(G), is the smallest cardinality of a restrained dominating set of G. A graph G is said to be cubic if every vertex has degree three. In this paper, we study restrained domination in cubic graphs. We show that if G is a cubic graph of order n, then

Some Generalized Bipartite Ramsey Numbers Involving Short Cycles

\gamma_{r}(G)\geq \frac{n}{4}

Restrained domination in unicyclic graphs

, and characterize the extremal graphs achieving this lower bound. Furthermore, we show that if G is a cubic graph of order n, then

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

\gamma _{r}(G)\leq \frac{5n}{11}.

Equality in a bound that relates the size and the restrained domination number of a graph

Lastly, we show that if G is a claw-free cubic graph, then γr(G)=γ(G).

An inequality that relates the size of a bipartite graph with its order and restrained domination number

For bipartite graphs

Nordhaus-Gaddum results for restrained domination and total restrained domination in graphs

G_1, G_2,\ldots ,G_k