A balanced tournament design, \(\mathrm{BTD}(n)\), defined on a \(2n\)-set \(V\), is an arrangement of the \(\binom{2n}{2}\) distinct unordered pairs of the elements of \(V\) into an \(n \times 2n-1\) array such that:
(1) every element of \(V\) is contained in precisely one cell of each column, and
(2) every element of \(V\) is contained in at most two cells of each row.

If we can partition the columns of a \(\mathrm{BTD}(n)\) defined on \(V\) into three sets \(C_1, C_2, C_3\) of sizes \(1, n-1, n-1\) respectively such that the columns in \(C_1 \cup C_2\) form a Howell design of side \(m\) and order \(2n\), an \(\mathrm{H}(n,2n)\), and the columns in \(C_1 \cup C_3\) form an \(\mathrm{H}(n,2n)\), then the \(\mathrm{BTD}(n)\) is called partitionable. We denote a partitioned balanced tournament design of side \(n\) by \(\mathrm{PBTD}(n)\). The existence of these designs has been determined except for seven possible exceptions. In this note, we describe constructions for four of these designs. This completes the spectrum of \(\mathrm{PBTD}(n)\) for \(n\) even.

In this note we complete the table of Ramsey numbers for \(K_s\) versus the family of all \((p,q)\)-graphs for \(p \leq 8\).
Moreover, some results are obtained for the general case.

Let \(G\) be a \(2\)-connected graph of order \(n\) with connectivity \(\kappa\) and independence number \(\alpha\). In this paper, we show that if for each independent set \(S\) with \(|S| = k+1\), there are \(u,v \in S\) such that satisfying one of the following conditions:

1. \(d(u) + d(v) \geq n\); or \(|N(u) \cap N(v)| \geq \alpha\); or \(|N(u) \cup N(v)| \geq n-k\);
2. for any \(x \in \{u,v\}\), \(y \in V(G)\) and \(d(x,y) = 2\), it implies that \(\max\{d(x), d(y)\} \geq n/2\),

then \(G\) is hamiltonian. This result reveals the internal relations among several well-known sufficient conditions: \((1)\) it shows that it does not need to consider all pairs of nonadjacent or distance two vertices in \(G\); \((2)\) it makes known that for different pairs of vertices in \(G\), it permits to satisfy different conditions.

Let \(G\) be a graph of order \(p\) such that both \(G\) and \(\overline{G}\) have no isolated vertices. Let \(\Upsilon_t\) and \(\overline{\Upsilon}_t\) denote respectively the total domination number of \(G\) and \(\overline{G}\). In this paper we obtain a characterization of all graphs \(G\) for which \\(i) \(\Upsilon_t +\overline{\Upsilon}_t= p+1\) and (ii) \(\Upsilon_t + \overline{\Upsilon}_t = p\).

The bondage number \(b(G)\) of a nonempty graph \(G\) was first introduced by Fink, Jacobson, Kinch, and Roberts [3]. In their paper they conjectured that \(b(G) \leq \Delta(G)+1\) for a nonempty graph \(G\). A counterexample for this conjecture was shown in [5]. Beyond it, we show now that there doesn’t exist an upper bound of the following form: \(b(G) \leq \Delta(G)+c\) for any \(c\in\mathbb{N}\).

It is shown that if \(t > 1\) and \(u \geq 5\), then the known necessary condition for the existence of a skew Room frame of type \(t^u\), is also sufficient with the possible exception of \((u, f)\) where \(u = 5\) and \(t \in \{11, 13, 17, 19, 23, 29, 31, 41, 43\}\).

The class of \(t-sc\) graphs constitutes a new generalization of self-complementary graphs. Many \(t-sc\) graphs exhibit a stable complementing permutation. In this paper, we prove a sufficient condition for the existence of a stable complementing permutation in a \(t-sc\) graph. We also construct several infinite classes of \(t-sc\) graphs to show the stringency of our sufficient condition.

A polyhex graph is either a hexagonal system or a coronoid system. A polyhex graph \(G\) is said to be \(k\)-coverable if for any \(k\) mutually disjoint hexagons the subgraph obtained from \(G\) by deleting all these \(k\) hexagons together with their incident edges has at least one perfect matching. In this paper, a constructive criterion is given to determine whether or not a given polyhex graph is \(k\)-coverable. Furthermore, a simple method is developed which allows us to determine whether or not there exists a \(k\)-coverable polyhex graph with exactly \(h\) hexagons.

A \((k, \lambda)\)-semiframe of type \(g^u\) is a group divisible design of type \(g^u\) \((\chi, \mathcal{G}, \mathcal{B})\), in which \(\mathcal{B}\) is written as a disjoint union \(\mathcal{B} = \mathcal{P} \cup \mathcal{Q} \) where \(\mathcal{P} \) is partitioned into partial parallel classes of \(\chi\) (with respect to some \(G \in \mathcal{G}\)) and \(\mathcal{Q} \) is partitioned into parallel classes of \(\chi\). In this paper, new constructions for these designs are provided with some series of designs with \(k = 3\). Cyclic semiframes are discussed. Finally, an application of semiframes is also mentioned.

A solution of Dudeney’s round table problem is given when \(n\) is as follows:

1. \(n = pq + 1\), where \(p\) and \(q\) are odd primes.
2. \(n = p^e + 1\), where \(p\) is an odd prime.
3. \(n = p^e q^f + 1\), where \(p\) and \(q\) are distinct odd primes satisfying \(p \geq 5\) and \(q \geq 11\), and \(e\) and \(f\) are natural numbers.