A new variation of the coloring problem, \(\mu\)-coloring, is defined in this paper. A coloring of a graph \(G = (V, E)\) is a function \(f: V \rightarrow \mathbb{N}\) such that \(f(v) \neq f(w)\) if \(v\) is adjacent to \(w\). Given a graph \(G = (V, E)\) and a function \(\gamma: V \rightarrow \mathbb{N}\), \(G\) is \(\mu\)-colorable if it admits a coloring \(f\) with \(f(v) \leq \mu(v)\) for each \(v \in V\). It is proved that \(\mu\)-coloring lies between coloring and list-coloring, in the sense of generalization of problems and computational complexity. Furthermore, the notion of perfection is extended to \(\mu\)-coloring, giving rise to a new characterization of cographs. Finally, a polynomial time algorithm to solve \(p\)-coloring for cographs is shown.

We introduce the notion of fuzzy \(K\)-ideals of \(K\)-algebras and investigate some of their properties. We characterize ascending and descending chains of \(K\)-ideals by the corresponding fuzzy \(K\)-ideals. We discuss some properties of characteristic fuzzy \(K\)-ideals of \(K\)-algebras. We construct a quotient \(K\)-algebra via fuzzy \(K\)-ideal and present the fuzzy isomorphism theorems.

Let \(P(G,\lambda)\) be the chromatic polynomial of a graph \(G\). A graph \(G\) is chromatically unique if for any graph \(H\), \(P(H,\lambda) = P(G, \lambda)\) implies \(H\) is isomorphic to \(G\). It is known that a complete tripartite graph \(K(a,b,c)\) with \(c \geq b \geq a \geq 2\) is chromatically unique if \(c – a \leq 3\). In this paper, we proved that a complete \(4\)-partite graph \(K(a,b,c,d)\) with \(d \geq c \geq b \geq a \geq 2\) is also chromatically unique if \(d – a \leq 3\).

In \([6]\), Cooperstein and Shult showed that the dual polar space \({DQ}^-(2n+1,\mathbb{K})\), \(\mathbb{K} = \mathbb{F}_q\), admits a full projective embedding into the projective space \({PG}(2^n – 1,\mathbb{K}’)\), \(\mathbb{K}’ = \mathbb{F}_{q^2}\). They also showed that this embedding is absolutely universal. The proof in \([6]\) makes use of counting arguments and group representation theory. Because of the use of counting arguments, the proof cannot be extended automatically to the infinite case. In this note, we shall give a different proof of their results, thus showing that their conclusions remain valid for infinite fields as well. We shall also show that the above-mentioned embedding of \({DQ}^-(2n + 1,\mathbb{K})\) into \({PG}(2^n -1,\mathbb{K}’)\) is polarized.

Let \(p\) be a prime number and let \(\mathbb{F}_p\) be a finite field. In the first section, we give some preliminaries from elliptic curves over finite fields. In the second section, we consider the rational points on the elliptic curves \(E_{p,\lambda} : y^2 = x(x-1)(x-\lambda)\) over \(\mathbb{F}_p\) for primes \(p \equiv 3 \pmod{4}\), where \(\lambda \neq 0, 1\). We prove that the order of \(E_{p,\lambda}\) over \(\mathbb{F}_p\) is \(p+1\) if \(\lambda = 2,\frac{p+1}{2}\) or \(p-1\). Later, we generalize this result to \(\mathbb{F}_{p^n}\) for any integer \(n \geq 2\). Also, we obtain some results concerning the sum of \(x\)- and \(y\)-coordinates of all rational points \((x,y)\) on \(E_{p,\lambda}\) over \(\mathbb{F}_p\). In the third section, we consider the rank of \(E_\lambda : y^2 = x(x-1)(x-\lambda)\) over \(\mathbb{Q}\).

For over a decade, there has been considerable research on codes over \(\mathbb{Z}_4\) and other rings. In spite of this, no tables or databases exist for codes over \(\mathbb{Z}_4\), as is the case with codes over finite fields. The purpose of this work is to contribute to the creation of such a database. We consider cyclic, negacyclic and quasi-twisted \((QT)\) codes over \(\mathbb{Z}_4\). Some of these codes have binary images with better parameters than the best-known binary linear codes. We call such codes “good codes”. Among these are two codes which improve the bounds on the best-known binary non-linear codes. Tables of best cyclic and \(QT\) codes over \(\mathbb{Z}_4\) are presented.

Acharya and Hegde have introduced the notion of strongly \(k\)-indexable graphs: A \((p,q)\)-graph \(G\) is said to be strongly \(k\)-indexable if its vertices can be assigned distinct integers \(0,1,2,\ldots,p-1\) so that the values of the edges, obtained as the sums of the numbers assigned to their end vertices can be arranged as an arithmetic progression \(k,k+1,k+2,\ldots,k+(q-1)\). Such an assignment is called a strongly \(k\)-indexable labeling of \(G\). Figueroa-Centeno et al. have introduced the concept of super edge-magic deficiency of graphs: Super edge-magic deficiency of a graph \(G\) is the minimum number of isolated vertices added to \(G\) so that the resulting graph is super edge-magic. They conjectured that the super edge-magic deficiency of the complete bipartite graph \(K_{m,n}\) is \((m-1)(n-1)\) and proved it for the case \(m=2\). In this paper, we prove that the conjecture is true for \(m=3,4,5\), using the concept of strongly \(k\)-indexable labelings \(^1\).