A set \(D\) of vertices in a graph \(G\) is a total dominating set if every vertex of \(G\) has at least one neighbor in \(D\). The minimum cardinality of a total dominating set of \(G\) is called the total domination number of \(G\), denoted by \(\gamma_t(G)\). A total dominating set of \(G\) with cardinality \(\gamma_t(G)\) is called a \(\gamma_t\)-set of \(G\). We characterize trees with unique \(\gamma_t\)-sets. Further, we prove that \(\gamma_t(G) \leq \frac{3}{5}n(G)\) for graphs with unique \(\gamma_t\)-sets, and we characterize all graphs with unique \(\gamma_t\)-sets where \(\gamma_t(G) = \frac{3}{5}n(G)\).

A word \(w = w_1w_2\ldots w_n\) avoids an adjacent pattern \(\tau\) iff \(w\) has no subsequence of adjacent letters having all the same pairwise comparisons as \(\tau\). In [12] and [13] the concept of words and permutations avoiding a single adjacent pattern was introduced. We investigate the probability that words and permutations of length \(n\) avoid two or three adjacent patterns.

We consider a variant of what is known as the discrete isoperimetric problem, namely the problem of minimising the size of the boundary of a family of subsets of a finite set. We use the technique of `shifting’ to provide an alternative proof of a result of Hart. This technique was introduced in the early \(1980s\) by Frankl and Füredi and gave alternative proofs of previously known classical results like the discrete isoperimetric problem itself and the Kruskal-Katona theorem. Hence our purpose is to bring Hart’s result into this general framework.

The domatic number of a graph \(G\) is the maximum number of dominating sets into which the vertex set of \(G\) can be partitioned.

We show that the domatic number of a random \(r\)-regular graph is almost surely at most \(r\), and that for \(3\)-regular random graphs, the domatic number is almost surely equal to \(3\).

We also give a lower bound on the domatic number of a graph in terms of order, minimum degree, and maximum degree. As a corollary, we obtain the result that the domatic number of an \(r\)-regular graph is at least \((r+1)/(3ln(r+1))\).

The concept of circular chromatic number of graphs was introduced by Vince \((1988)\). In this paper, we define the circular chromatic number of uniform hypergraphs and study their basic properties. We study the relationship between the circular chromatic number, chromatic number, and fractional chromatic number of uniform hypergraphs.

For a given Hadamard design \(D\) of order \(n\), we construct another Hadamard design \(D’\) of the same order, which is disjoint from \(D\).

The existence question for the family of \(4-(15,5,\lambda)\) designs has long been answered for all values of \(\lambda\) except \(\lambda = 2\). Here, we resolve this last undecided case and prove that \(4-(15, 5, 2)\) designs are constructible.

In this note, we prove that a graph is of class one if \(G\) can be embedded in a surface with positive characteristic and satisfies one of the following conditions:(i) \(\Delta(G) \geq 3\) and \(g(G)\)(the girth of \(G\)) \(\geq 8\) (ii) \(\Delta(G) \geq 4\) and \(g(G) \geq 5\)(iii) \(\Delta(G) \geq 5\) and \(g(G) \geq 4\).