We show that the double domination number of an \( n \)-vertex, isolate-free graph with minimum degree \( \delta \) is bounded above by \(\frac{n(\ln(\delta + 1) + \ln \delta + 1)}{\delta}.\) This result improves a previous bound obtained by J. Harant and M. A. Henning [On double domination in graphs, \({Discuss. Math. Graph Theory}\) \({25} (2005), 29-34]\). Further, we show that for fixed \( k \) and large \( \delta \), the \( k \)-tuple domination number is at most \(\frac{n(\ln \delta + (k – 1 + o(1))\ln \ln \delta)}{\delta},\) a bound that is essentially best possible.

Let \(\alpha\)-resolvable STS(\(v\)) denote a Steiner triple system of order \(v\) whose blocks are partitioned into classes such that each point of the design occurs in precisely \(\alpha\) blocks in each class. We show that for \(v \equiv u \equiv 1 \pmod{6}\) and \(v \geq 3u + 4\), there exists an \(\alpha\)-resolvable STS(\(v\)) containing an \(\alpha\)-resolvable sub-STS(\(u\)) for all suitable \(\alpha\).

A vertex set \( S \subseteq V(G) \) of a graph \( G \) is a \( 2 \)-dominating set of \( G \) if \( |N(v) \cap S| \geq 2 \) for every vertex \( v \in (V(G) – S) \), where \( N(v) \) is the neighborhood of \( v \). The \( 2 \)-domination number \( \gamma_2(G) \) of graph \( G \) is the minimum cardinality among the \( 2 \)-dominating sets of \( G \). In this paper, we present the following Nordhaus-Gaddum-type result for the \( 2 \)-domination number. If \( G \) is a graph of order \( n \), and \( \bar{G} \) is the complement of \( G \), then

\[ \gamma_2(G) + \gamma_2(\bar{G}) \leq n + 2, \]

and this bound is best possible in some sense.

The Graph Isomorphism (GI) problem asks if two graphs are isomorphic. Algorithms which solve GI have applications in, but not limited to, SAT solver engines, isomorph-free generation, combinatorial analysis, and analyzing chemical structures. However, no algorithm has been found which solves GI in polynomial time, implying that hard instances should exist. One of the most popular algorithms, implemented in the software package nauty, canonically labels a graph and outputs generators for its automorphism group. In this paper, we present some methods that improve its performance on graphs that are known to pose difficulty.

Let \( C \) be the set of distinct ways in which the vertices of a \( 5 \)-cycle may be coloured with at most two colours, called \({colouring\; types}\), and let \( S \subseteq C \). Suppose we colour the vertices of \( K_v \) with at most two colours. If \( \mathcal{D} \) is a \( 5 \)-cycle decomposition of \( K_v \), such that the colouring type of each \( 5 \)-cycle is in \( S \), and every colouring type in \( S \) is represented in \( \mathcal{D} \), then \( \mathcal{D} \) is said to have a \emph{proper colouring type} \( S \). For all \( S \) with \( |S| \leq 2 \), we determine some necessary conditions for the existence of a \( 5 \)-cycle decomposition of \( K_v \) with proper colouring type \( S \). In many cases, we show that these conditions are also sufficient.

Most computer algebra packages for Weyl groups use generators and relations and the Weyl group elements are expressed as reduced words in the generators. This representation is not unique and leads to computational problems. In [HHR06], the authors introduce the representation of Weyl group elements uniquely as signed permutations. This representation is useful for the study of symmetric spaces and their representations.

A computer algebra package enabling one to do computations related to symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. In this paper, we use the representation of Weyl group elements as signed permutations to improve the algorithms of [DH05]. These algorithms compute the fine structure of symmetric spaces and nice bases for local symmetric spaces.

A vertex subset \( X \) of a simple graph is called OC-irredundant (respectively CO-irredundant) if for each \( v \in X \), \( N(v) – N[X – \{v\}] \neq \emptyset \) (respectively \( N[v] – N(X – \{v\}) \neq \emptyset \)). Sharp bounds involving order and maximum degree for the minimum cardinality of a maximal OC-irredundant set and a maximal CO-irredundant set of a tree are obtained, and extremal trees are exhibited.