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.

Colbourn introduced \(V_\lambda(m, t)\) to construct transversal designs with index \(\lambda\). A \(V_\lambda(m, t)\) leads to a \((mt + 1, mt + 2; \lambda,0; t)\)-aussie-difference matrix. In this article, we use Weil’s theorem on character sums to show that for any integer \(\lambda \geq 2\), a \(V_\lambda(m, t)\) always exists in \(GF(mt + 1)\) for any prime power \(mt+1 > B_\lambda(m) = \left[\frac{E+\sqrt{E^2+4F}}{2}\right]^2\), where \(E = \lambda(u-1)(m-1)m^u-m^{u-1}+1,F=(u-1)\lambda m^u\) and \(u = \left\lfloor\frac{m{\lambda}+1+(-1)^{\lambda+1}}{2}\right\rfloor\). In particular, we determine the existence of \(V_{\lambda}(m, t)\) for \((\lambda, m) = (2, 2), (2, 3)\).

A weighted graph \((G,w)\) is a graph \(G = (V, E)\) together with a positive weight-function on its vertices \(w: V \to \mathbf{R}^{>0}\). The weighted domination number \(\gamma_w(G)\) of \((G, w)\) is the minimum weight \(w(D) = \sum_{v \in D} w(v)\) of a vertex set \(D \subseteq V\) with \(N[D] = V\), i.e. a dominating set of \(G\).

For this natural generalization of the well-known domination number, we study some of the classical questions of domination theory. We characterize all extremal graphs for the simple Ore-like bound \(\gamma_w(G) \leq \frac{1}{2}w(V)\) and prove Nordhaus-Gaddum-type inequalities for the weighted domination number.

Generalized Steiner systems GS\(_d(t,k,v,g)\) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size \(g + 1\) with minimum Hamming distance \(d\), in which each codeword has length \(v\) and weight \(k\). It was proved that the necessary conditions for the existence of a GS\(_4(2,4,v,g)\) are also sufficient for \(g = 2, 3\) and \(6\). In this paper, a general result on the existence of a GS\(_4(2,4,v,g)\) is presented. By using this result, we prove that the necessary conditions \(v \equiv 1 \pmod{3}\) and \(v \geq 7\) are also sufficient for the existence of a GS\(_4(2, 4, v, 4)\).

An \(A^3\)-code is an extension of an \(A\)-code in which none of the three participants, transmitter, receiver, and arbiter, is trusted. In this paper, we extend the previous model of \(A^3\)-codes by allowing the transmitter and the receiver to not only individually attack the system, but also collude with the arbiter against the other. We derive information-theoretic lower bounds on the success probability of various attacks, and combinatorial lower bounds on the size of key spaces. We also study the combinatorial structure of optimal \(A^3\)-codes against collusion attacks and give a construction of an optimal code.