For all nonnegative integers \(i,j\), let \(q(i, j)\) denote the number of all lattice paths in the plane from \((0,0)\) to \((i, j)\) with steps \((1,0)\), \((0,1)\), and \((1,1)\). In this paper, it is proved that

\[q(i_{n}p^{n}+…+i_0,j_np^n+…+j_0)\equiv(i_n,j_n)…q(i_0,j_0) \pmod{p}\]

where \(p\) is an odd prime and \(0 \leq i_k < p\), \(0 \leq j_k < p\). This relation implies a remarkable pattern to the divisibility of the array of numbers \(q(i, j)\).

Bauer and Tindell defined the graph invariant \(\wedge(G)\), for graphs \(G\) other than paths and the star \(K_{1,3}\), to be the least \(n\) for which \(G\) embeds in the \(n\)th iterated line graph of \(G\). They also proposed the problem of determining \(\wedge(T)\) for all trees \(T\). In this note, we completely solve this problem by showing that \(\wedge(T) = 3\) for any proper homeomorph \(T\) of \(K_{1,3}\) and that \(\wedge(T) = 2\) for all trees \(T\) which are neither paths nor homeomorphs of \(K_{1,3}\).

In a previous paper, all non-isomorphic decomposable \(3-(12,6,4)\) designs without repeated blocks were determined. These results are extended here by allowing repeated blocks. Under this condition, there are \(26\) non-isomorphic decomposable \(3-(12,6,4)\) designs, of which \(14\) have repeated blocks. Key blocks and point permutations for models of these designs are given, along with descriptions of their automorphism groups.

We extend the definition of edge-cordial graphs due to Ng and Lee for graphs on \(4k\), \(4k+1\), and \(4k+3\) vertices to include graphs on \(4k+2\) vertices, and show that, in fact, all graphs without isolated vertices are edge-cordial. Ng and Lee conjectured that all graphs on \(4k\), \(4k+1\), or \(4k+3\) vertices are edge-cordial.

We define the semibandwidth of a bipartite graph (whose bipartition is specified), which is a bipartite analogue of the bandwidth of a graph, and develop some of its properties. The motivation for this concept comes from the question of transforming a matrix by row and column permutations to as close to triangular form as possible.

Standard doubling and tripling constructions for block designs with block size three (triple systems) employ factorizations of complete graphs and of complete bipartite graphs. In these constructions, repeated edges in a factor lead to repeated blocks in the design. Hence the construction of triple systems with a prescribed number of repeated blocks is facilitated by determining the possible structure of repeated edges in the factors of a factorization of \(\lambda K_n\) and \(\lambda K_{n,n}\). For \(\lambda =3\), a complete determination of the possible combinations of numbers of doubly and triply repeated edges in 3-factorizations of \(\lambda K_n\) has been completed for \(n \geq12\). In this paper, we solve the analogous problem for the complete bipartite graphs in the case \(\lambda=3\). The case \(\lambda=1\) is trivial, and the case \(\lambda=2\) has been previously solved by Fu.

We obtain new base sequences, that is four sequences of lengths \(m + p\), \(m + p\), \(m\), \(m\), with \(p\) odd, which have zero auto correlation function which can be used with Yang numbers and four disjoint complementary sequences (and matrices) with zero non-periodic (periodic) auto correlation function to form longer sequences.

We give an alternate construction for \(T\)-sequences of length \((4n + 3)(2m + p)\), where \(n\) is the length of a Yang nice sequence.

These results are then used in the Goethals-Seidel or (Seberry) Wallis-Whiteman construction to determine eight possible decompositions into squares of \((4n + 3)(2m + p)\) in terms of the decomposition into squares of \(2m + 1\) when there are four suitable sequences of lengths \(m + 1\), \(m + 1\), \(m\), \(m\) and \(m\), the order of four Williamson type matrices. The new results thus obtained are tabulated giving \({OD}(4t; t, t, t, t)\) for the new orders \(t \in \{121, 135, 217, 221, 225, 231, 243, 245, 247,\)\( 253, 255, 259, 261, 265, 273,\) \(275, 279, 285, 287, 289, 295, 297, 299\}\).

The Hadamard matrix with greatest known excess for these new \(t\) is then listed.

We determine those pairs \((k,v)\), \(v = 4\cdot2^m, 5\cdot2^m\), for which there exists a pair of Steiner quadruple systems on the same \(v\)-set, such that the quadruples in one system containing a particular point are the same as those in the other system and moreover the two systems have exactly \(k\) other quadruples in common.

The point set “oval” has been considered in Steiner triple systems \((STS)\) and Steiner quadruple systems \((SQS)\) [3],[2]. There are many papers about “subsystems” in \(STS\) and \(SQS\). Generalizing and modifying the terms “oval” and “subsystem” we define the special point sets “near-oval” and “near-system” in Steiner quadruple systems. Considering some properties of these special point sets we specify how to construct \(SQS\) with near-ovals (\(S^{no}\)) and with near-systems (\(S^{ns}\)), respectively. For the same order of the starting system we obtain non-isomorphic systems \(S^{no}\) and \(S^{ns}\).

P. Paulraja recently showed that if every edge of a graph \(G\) lies in a cycle of length at most \(5\) and if \(G\) has no induced \(K_{i,s}\) as a subgraph, then \(G\) has a spanning closed trail. We use a weaker hypothesis to obtain a stronger conclusion. We also give a related sufficient condition for the existence of a closed trail in \(G\) that contains at least one end of each edge of \(G\).

Let \(G\) be a \(p\)-vertex graph which is rooted at \(x\). Informally, the rotation number \(h(G, x)\) is the smallest number of edges in a \(p\)-vertex graph \(F\) such that, for every vertex \(y\) of \(F\), there is a copy of \(G\) in \(F\) with \(x\) at \(y\). In this article, we consider rotation numbers for the generalized star graph consisting of \(k\) paths of length \(n\), all of which have a common endvertex, rooted at a vertex adjacent to the centre. In particular, if \(k = 3\) we determine the rotation numbers to within \(1, 2\) or \(3\) depending on the residue of \(n\) modulo \(6\).

The paper deals with combinatorial structures (pseudo-complexes, crystallizations) giving a direct link between the topology of triangulated manifolds and the theory of edge-coloured multigraphs. We define the concept of regular crystallization of a manifold and prove that every non-trivial handle-free closed \(n\)-manifold has a regular crystallization. Then we study some applications of regular crystallizations and give a counter-example to a conjecture of Y. Tsukui [20] about strong frames of the \(3\)-sphere.

A construction is given of a family of D-optimal designs of order \(n = 2v \equiv 2 \pmod{4}\), where \(v = 2q^2 + 2q + 1\) and \(q\) is an odd prime power. For \(q > 3\), all the orders of D-optimal designs produced by this construction are new.

The set of all distinct blocks of an \(t\)-(v,k) design is referred to as the support of the design, and its cardinality is denoted by \(b^*\). By generalizing a method on BIB designs called “trade off” to \(3\)-designs, a table for \(3\)-(9,4) designs with each \(60 \leq b^* \leq 126 = {\binom{9}{4}}\) is constructed. Also, we have produced over 2500 non-isomorphic \(3\)-(9,4) designs with \(\lambda = 6\). By utilizing this generalized trade off method along with two other methods, a table for \(3\)-(10,4) designs with 156 different \(b^*\)’s is constructed. By a recursive lower bound on the minimum value of \(b^*\) in all \(t\)-(v,k) designs, it is shown that \(b^*_{min}[3-(9,4)] \geq 36,\) and \(b^*_{min}[3\)-(10,4)] = 30.

A hypergraph has property \(\mathcal{B}\) (or chromatic number two) if there is a set which intersects each of its edges, but contains none of its edges. The number of edges in a smallest \(n\)-graph which does not have property \(\mathcal{B}\) is denoted \(m(n)\). This function has proved difficult to evaluate for \(n > 3\). As a consequence, several refinements and variations of the function \(m\) have been studied. In this paper, we describe an effort to construct, via computer, hypergraphs that improve current estimates of such functions.

We complete the construction of all the simple graphs with at most \(26\) vertices and transitive automorphism group. The transitive graphs with up to \(19\) vertices were earlier constructed by McKay , and the transitive graphs with \(24\) vertices by Praeger and Royle . Although most of the construction was done by computer, a substantial preparation was necessary. Some of this theory may be of independent interest.

Given a graph \(G\) and nonnegative integer \(k\), a map \(\pi: V(G) \to \{1, \ldots, k\}\) is a perfect \(k\)-colouring if the subgraph induced by each colour class is perfect. The perfect chromatic number of \(G\) is the least \(k\) for which \(G\) has a perfect \(k\)-colouring; such an invariant is a measure of a graph’s imperfection. We study here the theory of perfect colourings. In particular, the existence of perfect \(k\)-chromatic graphs are shown for all \(k\), and we draw attention to the associated extremal problem. We provide extensions to C. Berge’s Strong Perfect Graph Conjecture, and prove the existence of graphs with only one perfect \(k\)-colouring (up to a permutation of colours). The type of approach taken here can be applied to studying any graph property closed under induced subgraphs.

An \(S_{s,t}\) distar-factorization of \(DK_{m}\) is an edge partitioning of the complete symmetric directed graph \(DK_{m}\) into subdigraphs each of which is isomorphic to the distar \(S_{s,t}\) (the distar \(S_{s,t}\) being obtained from the star \(K_{1,s+t}\) by directing \(s\) of the edges into the centre and \(t\) of the edges out of the centre). We consider the question, “When can the arcs of \(DK_{m}\) be partitioned into arc-disjoint subgraphs each isomorphic to \(S_{s,t}\)?” and give necessary and sufficient conditions for \(S_{s,t}\) distar-factorizations of \(DK_{m}\) in the cases when either \(m\equiv 0\) or \(1 \pmod{s+t}\).