Let \(C\) be a plane convex body, and let \(l(ab)\) be the Euclidean length of a longest chord of \(C\) parallel to the segment \(ab\) in \(C\). By the relative length of \(ab\) in a convex body \(C\), we mean the ratio of the Euclidean length of \(ab\) to \(\frac{l(ab)}{2}\). We say that a side \(ab\) of a convex \(n\)-gon is relatively short if the relative length of \(ab\) is not greater than the relative length of a side of the regular \(n\)-gon. In this article, we provide a significant sufficient condition for a convex hexagon to have a relatively short side.

This paper studies families of self-orthogonal codes over \(\mathbb{Z}_4\). We show that the simplex codes (of Type \(a\) and Type \(\beta\)) are self-orthogonal. We answer the question of \(\mathbb{Z}_4\)-linearity for some codes obtained from projective planes of even order. A new family of self-orthogonal codes over \(\mathbb{Z}_4\) is constructed via projective planes of odd order. Properties such as self-orthogonality, weight distribution, etc. are studied. Finally, some self-orthogonal codes constructed from twistulant matrices are presented.

A complete paired comparison digraph \(D\) is a directed graph in which \(xy\) is an arc for all vertices \(x,y\) in \(D\), and to each arc we assign a real number \(0 \leq a \leq 1\) called a weight such that if \(xy\) has weight \(a\) then \(yx\) has weight \(1 – a\). We say that two vertices \(x, y\) dominate a third \(z\) if the weights on \(xz\) and \(yz\) sum to at least \(1\). If \(x\) and \(y\) dominate all other vertices in a complete paired comparison digraph, then we say they are a dominant pair. We construct the domination graph of a complete paired comparison digraph \(D\) on the same vertices as \(D\) with an edge between \(x\) and \(y\) if \(x\) and \(y\) form a dominant pair in \(D\). In this paper, we characterize connected domination graphs of complete paired comparison digraphs. We also characterize the domination graphs of complete paired comparison digraphs with no arc weight of \(.5\).

A graph \(G\) is a \((d,d+k)\)-graph, if the degree of each vertex of \(G\) is between \(d\) and \(d+k\). Let \(p > 0\) and \(d+k \geq 2\) be integers. If \(G\) is a \((d,d+k)\)-graph of order \(n\) with at most \(p\) odd components and without a matching \(M\) of size \(2|M| = n – p\), then we show in this paper that

1. \(n \geq 2d+p+2\) when \(p \leq k-2\),
2. \(n \geq 2\left\lceil \frac{d(p+2)}{k} \right\rceil +p +2\) when \(p \geq k-1\).

Corresponding results for \(0 \leq p \leq 1\) and \(0 \leq k \leq 1\) were given by Wallis \([6]\), Zhao \([8]\), and Volkmann \([5]\).

Examples will show that the given bounds (i) and (ii) are best possible.

In this paper, we prove that the cycle \(C_n\) with parallel chords and the cycle \(C_n\) with parallel \(P_k\)-chords are cordial for any odd positive integer \(k \geq 3\) and for all \(n \geq 4\) except for \(n \equiv 4r + 2, r \geq 1\). Further, we show that every even-multiple subdivision of any graph \(G\) is cordial and we show that every graph is a subgraph of a cordial graph.

A hypergraph is linear if no two distinct edges intersect in more than one vertex. A long standing conjecture of Erdős, Faber, and Lovász states that if a linear hypergraph has \(n\) edges, each of size \(n\), then its vertices can be properly colored with \(n\) colors. We prove the correctness of the conjecture for a new, infinite class of linear hypergraphs.

We use a computer to show that the crossing number of generalized Petersen graph \(P(10,3)\) is six.

Let \(G\) be a graph in which each vertex has been coloured using one of \(k\) colours, say \(c_1,c_2,\ldots,c_k\) If an \(m\)-cycle \(C\) in \(G\) has \(n_i\) vertices coloured \(c_i\), \(i = 1,2,\ldots,k\), and \(|n_i – n_j| \leq 1\) for any \(i,j \in \{1,2,\ldots,k\}\), then \(C\) is equitably \(k\)-coloured. An \(m\)-cycle decomposition \(C\) of a graph \(G\) is equitably \(k\)-colourable if the vertices of \(G\) can be coloured so that every \(m\)-cycle in \(C\) is equitably \(k\)-coloured. For \(m = 4,5\) and \(6\), we completely settle the existence problem for equitably \(2\)-colourable \(m\)-cycle decompositions of complete graphs and complete graphs with the edges of a \(1\)-factor removed.

Only the rotational tournament \(U_n\) for odd \(n \geq 5\), has the cycle \(C_n\) as its domination graph. To include an internal chord in \(C_n\), it is necessary for one or more arcs to be added to \(U_n\), in order to create the extended tournament \(U_n^+\). From this, the domination graph of \(U_t\), \(dom(U_n^+)\), may be constructed where \(C_k\), \(3 \leq k \leq n\), is a subgraph of \(dom(U_n^+)\). This paper explores the characteristics of the arcs added to \(U_n\) that are required to create an internal chord in \(C_n\).