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 \( v > 46 \).