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.

We derive several new lower bounds on the size of ternary covering codes of lengths \(6\), \(7\) and \(8\) and with covering radii \(2\) or \(3\).

We show that every complete graph \(K_n\), with an edge-colouring without monochromatic triangles, has a properly coloured Hamiltonian path.

In this paper we prove some basic properties of the \(g\)-centroid of a graph defined through \(g\)-convexity. We also prove that finding the \(g\)-centroid of a graph is NP-hard by reducing the problem of finding the maximum clique size of \(G\) to the \(g\)-centroidal problem. We give an \(O(n^2)\) algorithm for finding the \(g\)-centroid for maximal outer planar graphs, an \(O(m + n\log n)\) time algorithm for split graphs and an \(O(m^2)\) algorithm for ptolemaic graphs. For split graphs and ptolemaic graphs we show that the \(g\)-centroid is in fact a complete subgraph.