A binary linear code of length \(n\), dimension \(k\), and minimum distance at least \(d\) is called an \([n,k,d]\)-code. Let \(d(n,k) = \max \{d : \text{there exists an } [n,k,d]\text{-code}\}\). It is currently known by [6] that \(26 \leq d(66,13) \leq 28\). The nonexistence of a linear \([66,13,28]\)-code is proven.

In this paper, we completely solve the existence problem of \(\text{LOTS}(v)\) (i.e. large set of pairwise disjoint ordered triple systems of order \(v\)).

It is shown that a resolvable BIBD with block size five and index two exists whenever \(v \equiv 5 \pmod{10}\) and \(v \geq 50722395\). This result is based on an updated result on the existence of a BIBD with block size six and index unity, which leaves \(88\) unsolved cases. A construction using difference families to obtain resolvable BIBDs is also presented.

Functions \(c(n)\) and \(h(n)\) which count certain consecutive-integer partitions of a positive integer \(n\) are evaluated, and combinatorial interpretations of partitions with “\(c(n)\) copies of \(n\)” and “\(h(n)\) copies of \(n\)” are given.

J. Leech has posed the following problem: For each integer \(n\), what is the greatest integer \(N\) such that there exists a labelled tree with \(n\) nodes in which the distance between the pairs of nodes include the consecutive values \(1,2,\ldots,N\)? With the help of a computer, we get \(B(n)\) (the number \(N\) for branched trees) for \(2 \leq n \leq 10\) and lower bounds of \(B(11)\) and \(B(12)\). We also get \(U(n)\) (the number \(N\) for unbranched trees) for \(2 \leq n \leq 11\) independently, confirming some results gotten by J. Leech.

A method is presented for constructing simple partially balanced designs from \(t-(v,k,\lambda)\) designs. When the component designs satisfy a compatibility condition the result is a simple balanced design. The component designs can even be trivial (with some exceptions) with the resulting design being nontrivial. The automorphism group of the composition is given in terms of the automorphism groups of the component designs. Some previously unknown simple designs are constructed, including an infinite family of \(3\)-designs that are extremal with respect to an inequality of Cameron and Praeger. Some analogous theorems are given for difference families.

In this paper, constructions of simple cyclic \(2\)-designs are given. As a consequence, we determined the existence of simple \(2\)-\((q,k,\lambda)\) designs for every admissible parameter set \((q,k,\lambda)\) where \(q \leq 29\) is an odd prime power, with two undecided parameter sets \((q,k,\lambda) = (29,8,6)\) and \((29,8,10)\).

A map is an embedding of a graph into a surface so that each face is simply connected. Geometric duality, whereby vertices and faces are reversed, is a classic construction for maps. A generalization of map duality is given and discussed both graph and group theoretically.

We show how a claw-free well-covered graph containing no \(4\)-cycle, with any given independence number \(m\), can be constructed by linking together \(m\) sub-graphs, each isomorphic to either \(K_2\) or \(K_3\). We show further that the only well-covered connected claw-free graphs containing no \(4\)-cycle that cannot be constructed in this way are \(K_1\), and the cycle graphs on \(5\) and \(7\) vertices respectively.

In [3] R. Brauer asked the question: When is an \(n \times n\) complex matrix \(X\) the ordinary character table of some finite group? It is shown that the problem can be reduced in polynomial time to that of VERTEX INDEPENDENCE. We also pose and solve some (much) simpler problems of a related combinatorial nature.

Let \(\mathbb{F}_q = \text{GF}(q^n)\) denote the finite field of order \(q^n\), and let \(U_n = \cup_{i=1}^{n}(\mathbb{F}_i,)\). Explicit permutation-type formulas for the Frobenius map \(\varphi\) (defined by \((\varphi(x)) = x^q\)) on \(\mathbb{F}_n\) and on \(U_n\) are obtained by using the well-known number \(N_q\) (the number of monic irreducible polynomials of degree \(i\) in \(\mathbb{F}_1[x]\)). Some results in [1] and [2] can be obtained from these formulas. Moreover, some other results are also given by using Pólya’s counting theory.

The composition of two graphs \(G\) and \(H\), written \(G[H]\), is the graph with vertex set \(V(G) \times V(H)\) and \((u_1,v_1)\) is adjacent to \((u_2,v_2)\) if \(u_1\) is adjacent to \(u_2\) in \(G\) or if \(u_1 = u_2\) and \(v_1\) is adjacent to \(v_2\) in \(H\). The \(r\)th power of graph \(G\), denoted \(G^r\), is the graph with vertex set \(V(G)\) and edge set \(\{(u,v) : d(u,v) \leq r \text{ in } G\}\). The bandwidth of graph \(G\) is \(\min \max |f(u) – f(v)|\), where the max is taken over each edge \(uv \in E(G)\), and the min is over all proper numberings \(f\). This paper establishes tight upper and lower bounds for the bandwidth of an arbitrary graph composition \(G[H]\), with the upper bound based only on \(|V(H)|\) and the bandwidth of \(G\). In addition, the exact bandwidth of the composition of \(G[H]\) is established for \(G\) the power of a path or a cycle.

The edge-neighbor-connectivity of a graph \(G\) is the minimum size of all edge-cut-strategies of \(G\), where an edge-cut-strategy consists of a set of common edges of double stars whose removal disconnects the graph \(G\) or leaves a single vertex or \(\emptyset\). This paper discusses the extreme values of the edge-neighbor-connectivity of graphs relative to the connectivity, \(\kappa\), and gives two classes of graphs — one class with minimum edge-neighbor-connectivity, and the other one with maximum edge-neighbor-connectivity.

In \([V_2]\), Vince outlined three potential graph algorithms for \(S^3\) recognition, namely shelling, reducing, and closing; on the other hand, he conjectured that the graph \(H_0\ ) of Fig.1 – which proves the first two to fail – could be a counterexample for the third one, too. This note shows that the conjecture is false; so, the validity of the closing algorithm is still an open problem.

We consider two variations of the classical Ramsey number. In particular, we seek the number of vertices necessary to force the existence of an induced regular subgraph on a prescribed number of vertices.

The \(i\)-center \(C_i(G)\) of a graph \(G\) is the set of vertices whose distances from any vertex of \(G\) are at most \(i\). A vertex set \(X\) \(k\)-dominates a vertex set \(Y\) if for every \(y \in Y\) there is a \(x \in X\) such that \(d(x,y) \leq k\). In this paper, we prove that if \(G\) is a \(P_t\)-free graph and \(i \geq \lfloor\frac{t}{2}\rfloor \), then \(C_i(G)\) \((q+1)\)-dominates \(C_{i+q}(G)\), as conjectured by Favaron and Fouquet [4].

As a generalization of a matching consisting of edges only, Alavi et al. in [1] define a total matching which may contain both edges and vertices. Using total matchings, [1] defines a parameter \(\beta’_2(G)\) and proves that \(\beta’_2(G) \leq p-1\) holds for a connected graph of order \(p \geq 2\).
Our main result is to improve this inequality to \(\beta’_2(G) \leq p-2\sqrt{p}+{2}\) and we give an example demonstrating this bound to be best possible.
Relations of several other parameters to \(\beta’_2\) are demonstrated.

We examine permutations having a unique fixed point and a unique reflected point; such permutations correspond to permutation matrices having exactly one \(1\) on each of the two main diagonals. The permutations are of two types according to whether or not the fixed point is the same as the reflected point. We also consider permutations having no fixed or reflected points; these have been enumerated using two different methods, and we employ one of these to count permutations with unique fixed and reflected points.

Let \(G\) be a graph and \(t(G)\) be the number of triangles in \(G\). Define \(\mathcal G_n\) to be the set of all graphs on \(n\) vertices that do not contain a wheel and \(t_n = \max\{t(G) : G \in \mathcal G_n\}\).
T. Gallai conjectured that \(t_n \leq \lfloor\frac{n^2}{8}\rfloor\). In this note we describe a graph on \(n\) vertices that contains no wheel and has at least \(\frac{n^2+n}{8}-3\) triangles.

A construction is given which uses \(\text{PG}_i(d, q)\) and \(q\) copies of \(\text{AG}_i(d, q)\) to construct designs having the parameters of \(\text{PG}_{i+1}(d+1, q)\), where \(q\) is a prime power and \(i \leq d-1\).

In a previous paper, [6], we associated with every hyperoval of a projective plane of even order a Hadamard \(2\)–design and investigated when this design has lines with three points. We study further this problem using the concept of regular triple and prove the existence of lines with three points in Hadamard designs associated with translation hyperovals. In the general case, the existence of a secant line of regular triples implies that the order of the projective plane is a power of two.

An incomplete self-orthogonal latin square of order \(v\) with an empty subarray of order \(n\), an ISOLS(\(v, n\)), can exist only if \(v \geq 3n+1\). It is well known that an ISOLS(\(v, n\)) exists whenever \(v \geq 3n+1\) and \((v,n) \neq (6m+2,2m)\). In this paper we show that an ISOLS(\(6m+2, 2m\)) exists for any \(m \geq 24\).

Graceful and edge-graceful graph labelings are dual notions of each other in the sense that a graceful labeling of the vertices of a graph \(G\) induces a labeling of its edges, whereas an edge-graceful labeling of the edges of \(G\) induces a labeling of its vertices. In this paper we show a connection between these two notions, namely, that the graceful labeling of paths enables particular trees to be labeled edge-gracefully. Of primary concern in this enterprise are two conjectures: that a path can be labeled gracefully starting at any vertex label, and that all trees of odd order are edge-graceful. We give partial results for the first conjecture and extend the domain of trees known to be edge-graceful for the second conjecture.

In this paper we establish a number of new lower bounds on the size of a critical set in a latin square. In order to do this we first give two results which give critical sets for isotopic latin squares and conjugate latin squares. We then use these results to increase the known lower bound for specific classes of critical sets. Finally, we take a detailed look at a number of latin squares of small order. In some cases, we achieve an exact lower bound for the size of the minimal critical set.

We characterize “effectively” all greedy ordered sets, relative to the jump number problem, which contain no four-cycles. As a consequence, we shall prove that \(O(P) = G(P)\) \quad whenever \(P \text{ greedy ordered set with no four-cycles.}\)

This paper sketches the method of studying the Multiplier Conjecture that we presented in [1], and adds one lemma. Applying this method, we obtain some partial solutions for it: in the case \(v = 2n_1\), the Second Multiplier Theorem holds without the assumption ”\(n_1 > \lambda\)”, except for one case that is yet undecided where \(n_1\) is odd and \(7 \mid \mid v\) and \(t \equiv 3, 5,\) or \(6 \pmod{7}\), and for every prime divisor \(p (\neq 7)\) of \(v\) such that the order \(w\) of \(2\) mod \(p\) satisfies \(2|\frac {\phi(p)}{\omega}\); in the case \(n = 3n_1\) and \((v, 3 . 11) = 1\), then the Second Multiplier Theorem holds without the assumption “\(n_1 > \lambda\)” except for one case that is yet undecided where \(n_1\) cannot divide by \(3\) and \(13 \mid \mid v\) and the order of \(t\) mod \(13\) is \(12, 4,\) or \(6,2\), and for every prime divisor \(p (\neq 13)\) of \(v\) such that the order \(w\) of \(3\) mod \(p\) satisfies \(2|\frac {\phi(p)}{\omega}\). These results distinctly improve McFarland’s corresponding results and Turyn’s result.