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 \(g(G) \geq 4\).

In this paper, by using the generating function method, we obtain a series of identities involving the generalized Fibonacci and Lucas numbers.

This paper introduces the problem of finding a permutation \(\phi\) on the vertex set \(V(G)\) of a graph \(G\) such that the sum of the distances from each vertex to its image under \(\phi\) is maximized. We let \(\mathcal{S}(G) = \max \sum_{v\in V(G)} d(v, \phi(v))\), where the maximum is taken over all permutations \(\phi\) of \(V(G)\). Explicit formulae for several classes of graphs as well as general bounds are presented.

The local-edge-connectivity \((u,v)\) of two vertices \(u\) and \(v\) in a graph or digraph \(D\) is the maximum number of edge-disjoint \(u-v\) paths in \(D\), and the edge-connectivity of \(D\) is defined as \(\lambda(D) = \min\{\lambda(u, v) | u,v \in V(D)\}\). Clearly, \(\lambda(u,v) \leq \min\{d^+(u),d^-(v)\}\) for all pairs \(u\) and \(v\) of vertices in \(D\). We call a graph or digraph \(D\) maximally local-edge-connected when

\[\lambda(u, v) = \min\{d^+(u),d^-(v)\}\]

for all pairs \(u\) and \(v\) of vertices in \(D\).

Recently, Fricke, Oellermann, and Swart have shown that some known sufficient conditions that guarantee equality of \(\lambda(G)\) and minimum degree \(\delta(G)\) for a graph \(G\) are also sufficient to guarantee that \(G\) is maximally local-edge-connected.
In this paper we extend some results of Fricke, Oellermann, and Swart to digraphs and we present further sufficient conditions for
graphs and digraphs to be maximally local-edge-connected.

We show that every hamiltonian claw-free graph with a vertex \(x\) of degree \(d(x) \geq 7\) has a \(2\)-factor consisting of exactly two cycles.