Let \( c^* = \). If we remove the double edge, the result is a \( 4 \)-cycle. Let \( (S,T) \) be a \( 2 \)-fold triple system without repeated triples and \( (S,C^*) \) a pairing of the triples into copies of \( c^* \). If \( C \) is the collection of \( 4 \)-cycles obtained by removing the double edges from each copy of \( c^* \) and \( F \) is a reassembly of these double edges into \( 4 \)-cycles, then \( (S,C \cup F) \) is a \( 2 \)-fold \( 4 \)-cycle system. We show that the spectrum for \( 2 \)-fold triple systems having a \({metamorphosis}\) into a \( 2 \)-fold \( 4 \)-cycle system as described above is precisely the set of all \( n \equiv 0,1,4\, \text{or}\, 9 \pmod{12} \geq 5 \).

Consider a graph \( G \) in which the vertices are partitioned into \( k \) subsets. For each subset, we want a set of vertices of \( G \) that dominate that subset. Note that the vertices doing the domination need not be in the subset itself. We are interested in dominating the entire graph \( G \) as well as dominating each of the \( k \) subsets and minimizing the sum of these \( k + 1 \) dominating sets. For trees and for all values of \( k \), we can determine an upper bound on this sum and characterize the trees that achieve it.

A technique is described that constructs a 4-colouring of a planar triangulation in quadratic time. The method is based on iterating Kempe’s technique. The heuristic gives rise to an interesting family of graphs which cause the algorithm to cycle. The structure of these graphs is described. A modified algorithm that appears always to work is presented. These techniques may lead to a proof of the 4-Colour Theorem which does not require a computer to construct and colour irreducible configurations.

For \( k > 0 \), we call a graph \( G = (V,E) \) \( k \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_k^* \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_k \), defined by \(l^+(v) = \sum\{l(u,v): (u,v) \in E(G)\}\) is a constant map. We denote the set of all \( k \) such that \( G \) is \( k \)-magic by \(\text{IM}(G)\). We call this set the \textbf{\emph{integer-magic spectrum}} of \( G \). We investigate these sets for trees, double trees, and abbreviated double trees. We define group-magic spectrum for \( G \) similarly. Finally, we show that a tree is \( k \)-magic, \( k > 2 \), if and only if it is \( k \)-label reducible.

A graph \( G \) is 3-e.c. if for each distinct triple \( S \) of vertices, and each subset \( T \) of \( S \), there is a vertex not in \( S \) joined to the vertices of \( T \) and to no other vertices of \( S \). Few explicit examples of 3-e.c. graphs are known, although almost all graphs are 3-e.c. We provide new examples of 3-e.c. graphs arising as incidence graphs of partial planes resulting from affine planes. We also present a new graph operation that preserves the 3-e.c. property.

It is known that if a \( (22,33,12,8,4) \)-BIBD exists, then its incidence matrix is contained in a \( (33,16) \) doubly-even self-orthogonal code (that does not contain a coordinate of zeros). There are 594 such codes, up to equivalence. It has been theoretically proven that 116 of these codes cannot contain the incidence matrix of such a design. For the remaining 478 codes, an exhaustive clique search may be tried, on the weight 12 words of a code, to determine whether or not it contains such an incidence matrix. Thus far, such a search has been used to show 299 of the 478 remaining codes do not contain the incidence matrix of a \( (22,33,12,8,4) \)-BIBD.

In this paper, an outline of the method used to search the weight 12 words of these codes is given. The paper also gives estimations on the size of the search space for the remaining 179 codes. Special attention is paid to the toughest cases, namely the 11 codes that contain 0 weight 4 words and the 21 codes that contain one and only one weight 4 word.

Given a polyomino \( P \) with \( n \) cells, two players \( A \) and \( B \) alternately color the cells of the square tessellation of the plane. In the case of \( A \)-achievement, player \( A \) tries to achieve a copy of \( P \) in his color and player \( B \) tries to prevent \( A \) from achieving a copy of \( P \). The handicap number \( h(P) \) denotes the minimum number of cells such that a winning strategy exists for player \( A \). For all polyominoes that form a square of \( n = s^2 \) square cells, the handicap number will be determined to be \( s^2 – 1 \).

De Launey and Seberry have looked at the existence of Generalized Bhaskar Rao designs with block size 4 signed over elementary Abelian groups and shown that the necessary conditions for the existence of a \( (v, 4, \lambda; EA(g)) \) GBRD are sufficient for \( \lambda > g \) with 70 possible basic exceptions. This article extends that work by reducing those possible exceptions to just a \( (9, 4, 18h; EA(9h)) \) GBRD, where \( \gcd(6, h) = 1 \), and shows that for \( \lambda = g \) the necessary conditions are sufficient for \( v > 46 \).

If \(G\) and \(H\) are graphs, define the Ramsey number \(r(G, H)\) to be the least number \(p\) such that if the edges of the complete graph \(K_p\) are colored red and blue (say), either the red graph contains a copy of \(G\), or the blue graph contains a copy of \(H\). In this paper, we determine the Ramsey number \(r(mC_4, nC_5)\) for any \(m\geq1, n\geq1\).

We construct a complex \(K^n\) of \(m\)-ary relations, \(1 \leq m \leq n+1\), in a finite set \(X \neq Ø\), representing a model of an abstract cellular complex. For such a complex \(K^n\) we define the matrices of incidence and coincidence, the groups of homologies \(\mathcal{H}_m(K^n)\) and cohomologies \(\mathcal{H}^m(K^n)\) on the group of integers \(\mathbf{Z}\), and the Euler characteristic. On a combinatorial basis we derive their main properties. In further publications we will derive more analogues of classical properties, and also applications with respect to the existence of fixed relations in the utilization of the isomorphisms will be investigated. In particular, we intend to complete the theory of hypergraphs with the help of such topological observations.