In this paper, magic labelings of graphs are considered. These are labelings of the edges with integers such that the sum of the labels of incident edges is the same for all vertices. We particularly study positive magic labelings, where all labels are positive and different. A decomposition in terms of basis-graphs is described for such labelings. Basis-graphs are studied independently. A characterization of an algorithmic nature is given, leading to an integer linear programming problem. Some relations with other graph theoretical subjects, like vertex cycle covers, are discussed.

There are only two kinds of non-isomorphic consecutive vertex labelings of octahedron, and each of them can be deduced from the other. There is an algorithm to construct consecutive edge labelings. It is shown that there exist many non-isomorphic complementary consecutive edge labelings of octahedron.

It is known that there exists a one-to-one correspondence between the classes of equivalent \([n, n-k, 4]\)-codes over \(\mathrm{GF}(q)\) and the classes of projectively equivalent complete \(n\)-caps in \(\mathrm{PG}(k-1, q)\) (see [{20}], [{40}]). Hence all results on caps can be translated in terms of such codes. This fact stimulated many researches on the fundamental problem of determining the spectrum of the values of \(k\) for which there exist complete \(k\)-caps in \(\mathrm{PG}(n, q)\). This paper reports the result of a computer search for the spectrum of \(k\)’s that occur as a size of a complete \(k\)-cap in some finite projective spaces. The full catalog of such sizes \(k\) is given in the following projective spaces: \(\mathrm{PG}(3, q)\), for \(q \leq 5\), \(\mathrm{PG}(4, 2)\), \(\mathrm{PG}(4, 3)\), \(\mathrm{PG}(5, 2)\). Concrete examples of such caps are presented for each possible \(k\).\(^*\)

It is known (cf. {Hamada} [12] and {BrouwerEupen} and van Eupen [2] ) that (1) there is no ternary \([230, 6, 153]\) code meeting the Griesmer bound but (2) there exists a ternary \([232, 6, 153]\) code. This implies that \(n_3(6, 153) = 231\) or \(232\), where \(n_3(k, d)\) denotes the smallest value of \(n\) for which there exists a ternary \([n, k, d]\) code. The purpose of this paper is to prove that \(n_3(6, 153) = 232\) by proving the nonexistence of ternary \([231, 6, 153]\) codes.

If \(D\) is an acyclic digraph, its competition graph is an undirected graph with the same vertex set and an edge between vertices \(x\) and \(y\) if there is a vertex \(a\) so that \((x,a)\) and \((y,a)\) are both arcs of \(D\). If \(G\) is any graph, \(G\) together with sufficiently many isolated vertices is a competition graph, and the competition number of \(G\) is the smallest number of such isolated vertices. Roberts \([1978]\) gives an elimination procedure for estimating the competition number and Opsut \([1982]\) showed that this procedure could overestimate. In this paper, we modify that elimination procedure and then show that for a large class of graphs it calculates the competition number exactly.

A new concept of genus for finite groups, called stiff genus, is developed. Cases of stiff embeddings in orientable or nonorientable surfaces are dealt with. Computations of stiff genus of several classes of abelian and non-abelian groups are presented. A comparative analysis between the stiff genus and the Tucker symmetric genus is also undertaken.

For each admissible \(v\) we exhibit a \(\mathrm{H}(v, 3, 1)\) with a spanning set of minimum cardinality and a \(\mathrm{H}(v, 3, 1)\) with a scattering set of maximum cardinality.

Using the Jacobi triple product identity and the quintuple product identity, we obtain identities involving several partition functions.

A snark is a simple, cyclically \(4\)-edge connected, cubic graph with girth at least \(5\) and chromatic index \(4\). We give a complete list of all snarks of order less than \(30\). Motivated by the long standing discussion on trivial snarks (i.e. snarks which are reducible), we also give a brief survey on different reduction methods for snarks. For all these reductions we give the complete numbers of irreducible snarks of order less than \(30\) and the number of nonisomorphic \(3\)-critical subgraphs of these graphs. The results are obtained with the aid of a computer.

We give short proofs of theorems of Nash-Williams (on edge-partitioning a graph into acyclic subgraphs) and of Tutte (on edge-partitioning a graph into connected subgraphs). We also show that each theorem can be easily derived from the other.