Let \( a, b \) be two positive integers. For the graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \) with \( p = |V(G)| \) and \( q = |E(G)| \), we define two sets \( Q(a) \) and \( P(b) \) as follows:

\[
Q(a) =
\begin{cases}
\{\pm a, \pm(a+1), \ldots, \pm(a+\frac{q-2}{2})\} & \text{if } q \text{ is even} \\
\{0\} \cup \{\pm a, \pm(a+1), \ldots, \pm(a + (q-3)/{2})\} & \text{if } q \text{ is odd}
\end{cases}
\]

\[
P(b) =
\begin{cases}
\{\pm b, \pm(b+1), \ldots, \pm(b + (p-2)/{2})\} & \text{if } p \text{ is even} \\
\{0\} \cup \{\pm b, \pm(b+1), \ldots, \pm(b + (\frac{p-3}{2})/2)\} & \text{if } p \text{ is odd}
\end{cases}
\]

For the graph \( G \) with \( p = |V(G)| \) and \( q = |E(G)| \), \( G \) is said to be \( Q(a)P(b) \)-super edge-graceful (in short \( Q(a)P(b) \)-SEG), if there exists a function pair \( (f, f^+) \) which assigns integer labels to the vertices and edges; that is, \( f^+ : V(G) \to P(b) \), and \( f: E(G) \to Q(a) \) such that \( f^+ \) is onto \( P(b) \) and \( f \) is onto \( Q(a) \), and

\[
f^+(u) = \sum\{f(u,v) : (u,v) \in E(G)\}.
\]

We investigate \( Q(a)P(b) \) super edge-graceful graphs.

Let \( A \) be a non-trivial abelian group. We call a graph \( G = (V,E) \) \( A \)-magic if there exists a labeling \( f : E(G) \to A \setminus \{0\} \) such that the induced vertex set labeling \( f^+ : V(G) \to A \), defined by \( f^+(v) = \sum f(u,v) \) where the sum is over all \( (u,v) \in E(G) \), is a constant map. In this paper, we show that \( K_{k_1,k_2,\ldots,k_n} \) (where \( K_{i} \geq 2 \)) is \( A \)-magic, for all \( A \) where \( |A| \geq 3 \).

We define 1 new type of resolvability called \( \alpha \)-pair-resolvability in which each point appears in each resolution class as a member of \( \alpha \)-pairs. The concept is intended for path designs (or other designs) in which the role of points in blocks is not uniform or for designs which are not balanced. We determine the necessary conditions and show they are sufficient for \( k = 3 \) and \( \alpha = 2,3 \) (\( \alpha \geq 2 \) is necessary in every case). We also consider near \( a \)-pair-resolvability and show the necessary conditions are sufficient for \( \alpha = 2,4 \). We consider under what conditions it is possible for the ordered blocks of a path design to be considered as unordered blocks and thereby create a triple system (a tight embedding) and there also we show the necessary conditions are sufficient. We show it is always possible to embed maximally unbalanced path designs \( \text{PATH}(v, 3, 1) \) into \( \text{PATH}(v + s, 3, 1) \) for admissible \( s \), and to embed any \( \text{PATH}(v, 3, 2\lambda) \) into a \( \text{PATH}(v + s,3, 2\lambda) \) for any \( s \geq 1 \).

Recent developments in logic programming are based on bilattices (algebras with two separate lattice structures). This paper provides characterizations and structural descriptions for bilattices using the algebraic concepts of superproduct and hyperidentity. The main structural description subsumes the many variants that have appeared in the literature.

The Ramsey number \( R(C_4, B_n) \) is the smallest positive integer \( m \) such that for every graph \( F \) of order \( m \), either \( F \) contains \( C_4 \) (a quadrilateral) or \( \overline{F} \) contains \( B_n \) (a book graph \( K_2 + \overline{K_n} \) of order \( n+2 \)). Previously, we computed \( R(C_4, B_n) = n+9 \) for \( 8 \leq n \leq 12 \). In this continuing work, we find that \( R(C_4, B_{13}) = 22 \) and surprisingly \( R(C_4, B_{14}) = 24 \), showing that their values are not incremented by one, as one might have suspected. The results are based on computer algorithms.

Comma-free codes are used to correct synchronization errors in sequential transmission. Systematic comma-free codes have codewords with fixed positions for error correction. We consider only comma-free codes with constant word length \( n > 1 \). Circular codes use the integers mod \( n \) as indices for codeword entries. We first show two easily stated conditions are equivalent to the existence question for circular systematic comma-free codes over arbitrary finite alphabets. For \( n > 3 \) a family of circular systematic comma-free codes with word length \( n = p \), a prime, is constructed, each corresponding to a fair partition of a difference set in \( \mathbb{Z}_n \).

This paper gives the exact size of edit spheres of radius 1 and 2 for any word over a finite alphabet. Structural information about the edit metric space, in particular a representation as a pyramid of hypercubes, will be given. The 1-spheres are easy to understand, being identical to 1-spheres over the Hamming metric. Edit metric 2-spheres are much more complicated. The size of a 2-sphere hinges on the structure of the word at its center. That is, the word’s length, number of blocks, and most importantly (and troublesome) the number of locally maximal alternating substrings (LMAS) of each length. An alternating substring switches back and forth between two characters, e.g. 010101, and is maximal if it is contained in no other such substring. This variation in sphere size depending on center characteristics is what truly separates the algebraic character of codes over the edit metric from those over the Hamming metric.

We determine some coefficients of the flow polynomial of the complete graph \( K_n \).

Groups provide the mathematical language for exact symmetry. Applications in biology and other fields are now raising the problem of developing a rigorous theory of approximate symmetry. In this paper, it is shown how approximate symmetry is determined by a quasigroup.

A graph has the neighbour-closed-co-neighbour, or ncc property, if for each of its vertices \(x\), the subgraph induced by the neighbour set of \(x\) is isomorphic to the subgraph induced by the closed non-neighbour set of \(x\). Graphs with the ncc property were characterized in [1] by the existence of a locally \(C_4\) perfect matching \(M\): every two edges of \(M\) induce a subgraph isomorphic to \(C_4\). In the present article, we investigate variants of locally \(C_4\) perfect matchings. We consider the cases where pairs of distinct edges of the matching induce isomorphism types including \(P_4\), the paw, or the diamond. We give several characterizations of graphs with such matchings. In addition, we supply characterizations of graphs with matchings whose edges satisfy a prescribed parity condition.