In this paper, it has been proved that \(K_{r,r} \times K_{m}\), \(m \geq 3\), is hamiltonian decomposable.

A twofold extended triple system with two idempotent elements, \(TETS(v)\), is a pair \((V, B)\), where \(V\) is a \(v\)-set and \(B\) is a collection of triples, called blocks, of type \(\{x,y,z\}\), \(\{x,x,y\}\) or \(\{x,x,x\}\) such that every pair of elements of \(V\), not necessarily distinct, belongs to exactly two triples and there are only two triples of the type \(\{x, x, x\}\).
This paper shows that an indecomposable \(TETS(v)\) exists which contains exactly \(k\) pairs of repeated blocks if and only if \(v \not\equiv 0 \mod 3\), \(v \geq 5\) and \(0 \leq k \leq b_v – 2\), where \(b_v = \frac{(v + 2)(v + 1)}{6}\).

For a subset of vertices \(S\) in a graph \(G\), if \(v \in S\) and \(w \in V-S\), then the vertex \(w\) is an \(external\; private\; neighbor\; of \;v\) (with respect to \(S\)) if the only neighbor of \(w\) in \(S\) is \(v\). A dominating set \(S\) is a private dominating set if each \(v \in S\) has an external private neighbor. Bollébas and Cockayne (Graph theoretic parameters concerning domination, independence and irredundance. J. Graph Theory \(3 (1979) 241-250)\) showed that every graph without isolated vertices has a minimum dominating set which is also a private dominating set. We define a graph \(G\) to be a \(private\; domination\; graph\) if every minimum dominating set of \(G\) is a private dominating set. We give a constructive characterization of private domination trees.

The Levi graph of a balanced incomplete block design is the bipartite graph whose vertices are the points and blocks of the design, with each block adjacent to those points it contains. We derive upper and lower bounds on the isoperimetric numbers of such graphs, with particular attention to the special cases of finite projective planes and Hadamard designs.

This note proves that, given one member, \(T\), of a particular family of radius-three trees, every radius-two, triangle-free graph, \(G\), with large enough chromatic number contains an induced copy of \(T\).

Let \(k(D)\) be the index of convergence of a digraph \(D\) of order \(n \geq 8\). It is proved that if \(D\) is not strong with only minimally strong components and the greatest common divisor of the cycle lengths of \(D\) is at least two, then

\[k(D) \leq \begin{cases}
\frac{1}{2}(n^2 – 8n + 24) & \text{if } n \text{ is even}, \\
\frac{1}{2}(n^2 – 10n + 35) & \text{if } n \text{ is odd}.
\end{cases}\]

The cases of equality are also characterized.

Let \(C_n\) denote the cycle with \(n\) vertices, and \(C_n^{(t)}\) denote the graphs consisting of \(t\) copies of \(C_n\), with a vertex in common. Koh et al. conjectured that the graphs \(C_n^{(t)}\) are graceful if and only if \(nt \equiv 0, 3 \pmod{4}\). The conjecture has been shown true for \(n = 3, 5, 6, 4k\). In this paper, the conjecture is shown to be true for \(n = 7\).

It is well known that a linear code over a finite field with the systematic generator matrix \([I | P]\) is MDS (Maximum Distance Separable) if and only if every square submatrix of \(P\) is nonsingular. In this correspondence, we obtain a similar characterization for the class of Near-MDS codes in terms of the submatrices of \(P\).

In this paper, we define the signed total domatic number of a graph in an analogous way to that of the fractional domatic number defined by Rall (A fractional version of domatic number. Congr. Numer. \(74 (1990), 100-106)\). A function \(f: V(G) \to \{-1,1\}\) defined on the vertices of a graph \(G\) is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A set \(\{f_1,\ldots,f_a\}\) of signed total dominating functions on \(G\) such that \(\sum\limits_{i=1}^a f_i(v) \leq 1\) for each vertex \(v \in V(G)\) is called a signed total dominating family of functions on \(G\). The signed total domatic number of \(G\) is the maximum number of functions in a signed total dominating family of \(G\). In this paper, we investigate the signed total domatic number for special classes of graphs.

Let \(G\) be the product of two directed cycles, let \(Z_a\) be a subgroup of \(Z_a\), and let \(Z_d\) be a subgroup of \(Z_b\). Also, let \(A = \frac{a}{c}\) and \(B = \frac{b}{d}\). We say that \(G\) is \((Z_c \times Z_d)\)-hyperhamiltonian if there is a spanning connected subgraph of \(G\) that has degree \((2, 2)\) at the vertices of \(Z_c \times Z_d\) and degree \((1, 1)\) everywhere else. We show that the graph \(G\) is \((Z_c \times Z_d)\)-hyperhamiltonian if and only if there exist positive integers \(m\) and \(n\) such that \(Am + Bn = AB + 1\), \(gcd(m, n) = 1\) or \(2\), and when \(gcd(m, n) = 2\), then \(gcd(dm, cn) = 2\).