A critical set in a Latin square of order \(n\) is a set of entries in a Latin square which can be embedded in precisely one Latin square of order \(n\). Also, if any element of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order \(n\). In this paper, we find smallest weak and smallest totally weak critical sets for all the Latin squares of orders six and seven. Moreover, we computationally prove that there is no (totally) weak critical set in the back circulant Latin square of order five and we find a totally weak critical set of size seven in the other main class of Latin squares of order five.

In this paper, we give the following labelings:

1. Elegant labelings of triangular snakes \(\Delta_{n}\) , \(n \equiv 0,1,2 \mod 4\).
2. Near-elegant labeling of triangular snakes \(\Delta_{n}\) when \(n \equiv 3 \mod 4\), which are not elegant.
3. Elegant and near-elegant labelings of some of the theta graphs \(\theta_{n,n}\) when \(n = 1, 2, 3\).
4. Harmonious labelings of helms \(H_n\) when \(n\) is even.

A linear \([n,k,d]_q\) code \(C\) is called NMDS if \(d(C) = n – k\) and \(d(C^{\perp}) = k\). In this paper, the classification of the \([n,3,n-k]_q\) NMDS codes is given for \(q = 7,8,9\). It has been found using the correspondence between \([n,3,n-k]_q\) NMDS codes and \((n,3)\)-arcs of \(\mathrm{PG}(2,q)\).

A path in a digraph is antidirected if the two adjacent edges of the path have opposing orientations. In this paper, we give a necessary and sufficient condition for the edges of the complete symmetric graph to be decomposed into isomorphic antidirected paths.

The aim of this note is to provide a programme for the Computer Algebra package MAGMA, which is suitable to decode one-point Goppa codes defined from Hermitian curves.

In this article, the intersection problem for twin bowtie and near bowtie systems is completely solved.

Given a graph, a no-hole \(2\)-distant coloring (also called \(N\)-coloring) is a function \(f\) that assigns to each vertex a non-negative integer (color) such that the separation of the colors of any pair of adjacent vertices must be at least \(2\), and all the colors used by \(f\) form a consecutive set (the no-hole assumption). The minimum consecutive \(N\)-span of \(G\), \(csp(G)\), is the minimum difference of the largest and the smallest colors used in an \(N\)-coloring of \(G\), if there exists such a coloring; otherwise, define \(csp(G) = \infty\). Here we investigate the exact values of \(csp(G)\) for unit interval graphs (also known as \(1\)-unit sphere graphs). Earlier results by Roberts [18] indicate that if \(G\) is a unit interval graph on \(n\) vertices, then \(csp_1(G)\) is either \(2\chi(G) – 1\) or \(2\chi(G) – 2\), if \(n > 2\chi(G) – 1\); \(csp_1(G) = \infty\), if \(n < 2\chi(G) – 1\), where \(\chi(G)\) denotes the chromatic number. We show that in the former case (when \(n > 2\chi(G) – 1\)), both values of \(csp_1(G)\) are attained, and give several families of unit interval graphs such that \(csp_1(G) = 2\chi(G) – 2\). In addition, the exact values of \(csp_1(G)\) are completely determined for unit interval graphs with \(\chi(G) = 3\).

Let \(G\) be a graph. Let \(\gamma\) denote the minimum cardinality of a dominating set in \(G\). Let \(\beta\), respectively \(i\), denote the maximum, respectively minimum, cardinality of a maximal independent set in \(G\). We show \(\gamma + \Delta \geq \left\lceil {2\sqrt{n}-1} \right\rceil\), where \(n\) is the number of vertices of \(G\). A straightforward construction shows that given any \(G’\) there exists a graph \(G\) such that \(\gamma(G) + \Delta(G) = \left\lceil {2\sqrt{n}-1} \right\rceil\) and \(G’\) is an induced subgraph of \(G\), making classification of these \(\gamma+\Delta\) minimum graphs difficult.

We then focus on the subclass of these graphs with the stronger condition that \(\beta + \Delta = \left\lceil {2\sqrt{n}-1} \right\rceil\). For such graphs \(i = \beta\) and thus the graphs are well-covered. If \(G\) is a graph with \(\beta + \Delta = \left\lceil {2\sqrt{n}-1} \right\rceil\), we have \(\beta = \left\lceil \frac{\sqrt{n}}{\Delta+1} \right\rceil\). We give a catalogue of all well-covered graphs with \(\Delta \leq 3\) and \(\beta = \left\lceil \frac{\sqrt{n}}{\Delta+1} \right\rceil\). Again we establish that given any \(G’\) we can construct \(G\) such that \(G’\) is an induced subgraph of \(G\) and \(G\) satisfies \(\beta = \left\lceil \frac{\sqrt{n}}{\Delta+1} \right\rceil\). In fact, the graph \(G\) can be constructed so that \(\beta(G) + \Delta(G) = \left\lceil {2\sqrt{n}-1} \right\rceil\). We remark that \(\Delta(G)\) may be much larger than \(\Delta(G’)\).

We conclude the paper by analyzing integer solutions to \(\left\lceil \frac{n}{\Delta+1} \right\rceil + \Delta = \left\lceil {2\sqrt{n}-1} \right\rceil\). In particular, for each \(n\), the values of \(\Delta\) that satisfy the equation form an interval. When \(n\) is a perfect square, this interval contains only one value, namely \(\sqrt{n}\). For each \((n, \Delta)\) solution to the equation, there exists a graph \(G\) with \(n\) vertices, maximum degree \(\Delta\), and \(\beta = \left\lceil \frac{\sqrt{n}}{\Delta+1} \right\rceil\).

We construct a family of \(p-1\) square \(p \times p\) matrices (\(p\) is any prime) whose periodic cross-correlation values are uniformly \(-p, 0, +p\) between all pairs of the matrices in the family. For every one of the matrices in the family, all the off-peak autocorrelation values are \(-p\) and \(0\), while the single peak value is \(p(p-1)\). For \(p = 127\) (where the values \(-p, 0, +p\) are below \(1\%\) of the size \(p^2\) of the matrices) utilization of this construction has resulted in the superimposed embedding of twelve of the matrices (as watermarks) in the standard image “Lenna” and their subsequent retrieval without recourse to the unmarked image.