We determine upper bounds for $\gamma \left(Q{n}^t\right)$ and $i\left(Q{ₙ}^t\right)$, the domination and independent domination numbers, respectively, of the graph $Q{ₙ}^t$ obtained from the moves of queens on the n×n chessboard drawn on the torus.

For a graph property $𝒫$ and a graph $G$, we define the domination subdivision number with respect to the property $𝒫$ to be the minimum number of edges that must be subdivided (where each edge in $G$ can be subdivided at most once) in order to change the domination number with respect to the property $𝒫$. In this paper we obtain upper bounds in terms of maximum degree and orientable/non-orientable genus for the domination subdivision number with respect to an induced-hereditary property, total domination...

A b-coloring of a graph G by k colors is a proper vertex coloring such that every color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k-1 color classes. The b-chromatic number ${\chi }_b\left(G\right)$ is the maximum integer k for which G has a b-coloring by k colors. Moreover, the graph G is called b-continuous if G admits a b-coloring by k colors for all k satisfying $\chi \left(G\right)\le k\le {\chi }_b\left(G\right)$. In this paper, we establish four general upper bounds on ${\chi }_b\left(G\right)$. We present results on the b-chromatic number...

Let G be a graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and Σx∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that Σdi=1 fi(x) ≤ k for each x ∈ V (G), is called a signed total (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed...

The oriented chromatic number of an oriented graph ${}^{\to }G$ is the minimum order of an oriented graph ${}^{\to }H$ such that ${}^{\to }G$ admits a homomorphism to ${}^{\to }H$. The oriented chromatic number of an undirected graph G is then the greatest oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper oriented chromatic number of an undirected graph G, defined as the minimum order of an oriented graph ${}^{\to }U$ such that every orientation ${}^{\to }G$ of G admits a homomorphism to ${}^{\to }U$. We give some properties...

In this paper we build and analyze networks using the statistical and programming environment R and the igraph package. We investigate random, small-world, and scale-free networks and test a standard problem of connectivity on a random graph. We then develop a method to study how vaccination can alter the structure of a disease transmission network. We also discuss a variety of other uses for networks in biology.

Dans cet article, nous utilisons un paramètre $\sigma \left(T\right)$ défini à partir des scores d’un tournoi $T$ pour déterminer les ordres médians de $T$. Ce paramètre évalue un éloignement entre le tournoi $T$ et les tournois transitifs ayant le même nombre de sommets. Appelant $i\left(T\right)$ le nombre minimum d’arcs à inverser pour rendre $T$ transitif, et $n$ le nombre de sommets de $T$, nous proposons d’abord deux algorithmes linéaires en n calculant $i\left(T\right)$ et un ordre médian de $T$ pour les tournois $T$ tels que $\sigma \left(T\right)$ soit égal à $1$ ou $2$. Puis nous...