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 \(^1\).

Let \(M = \{v_1, v_2, \ldots, v_t\}\) be an ordered set of vertices in a graph \(G\). Then \((d(u, v_1), d(u, v_2), \ldots, d(u, v_\ell))\) is called the \(M\)-location of a vertex \(u\) of \(G\). The set \(M\) is called a locating set if the vertices of \(G\) have distinct \(M\)-locations. A minimum locating set is a set \(M\) with minimum cardinality. The cardinality of a minimum locating set of \(G\) is called the Location Number \(L(G)\). This concept has wide applications in motion planning and in the field of robotics. In this paper, we consider networks with a binary tree as an underlying structure and determine the minimum locating set of such architectures. We show that the location number of an \(n\)-level \(X\)-tree lies between \(2^{n-3}\) and \(2^{n – 3} + 2\). We further prove that the location number of an \(N \times N\) mesh of trees is greater than or equal to \(N/2\) and less than or equal to \(N\).

In this paper, we give generalizations of Padovan numbers and Perrin numbers. We apply these generalizations for counting of special subsets of the set of \(n\) integers. Next, we give their graph representations with respect to the number of maximal \(k\)-independent sets in graphs.

In this paper, we show that the crossing number of the complete multipartite graph \(K_{1,1,3,n}\) is

\[\operatorname{cr}(K_{1,1,3,n}) = 4\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor + \lfloor\frac{3n}{2}\rfloor\]

Our proof depends on Kleitman’s results for the complete bipartite graphs [D. J. Kleitman, The crossing number of \(K_{5,n}\), J. Combin.Theory, \(9 (1970), 315-323\)]..

A near-perfect matching is a matching saturating all but one vertex in a graph. In this note, it is proved that if a graph has a near-perfect matching then it has at least two, moreover, a concise structure construction for all graphs with exactly two near-perfect matchings is given. We also prove that every connected claw-free graph \(G\) of odd order \(n\) (\(n \geq 3\)) has at least \(\frac{n+1}{2}\) near-perfect matchings which miss different vertices of \(G\).

In this paper, we introduce some contractive conditions of Meir-Keeler type for a pair of mappings, called MK-pair and L-pair, in the framework of cone metric spaces. We prove theorems which assure the existence and uniqueness of common fixed points for MK-pairs and L-pairs. As an application, we obtain a result on the common fixed point of a p-MK-pair, a mapping, and a multifunction in complete cone metric spaces. These results extend and generalize well-known comparable results in the literature.

Four new combinatorial identities involving certain generalized \(F\)-partition functions and \(n\)-colour partition functions are proved bijectively. This leads to new combinatorial interpretations of four mock theta functions of S.Ramanujan.

le of an edge-coloured graph \(G^*\) such that there is no finite integer \(n\) for which it is possible to decompose \(rK_n^*\) into edge-disjoint colour-identical copies of \(G^*\). We investigate the problem of determining precisely when an edge-coloured graph \(G^*\) with \(r\) colours admits a \(G^*\)-decomposition of \(rK_n^*\), for some finite \(n\). We also investigate conditions under which any partial edge-coloured \(G^*\)-decomposition of \(rK_n^*\) has a finite embedding.

Let \(G\) be a connected graph, and let \(d(u,v)\) denote the distance between vertices \(u\) and \(v\) in \(G\). For any cyclic ordering \(\pi\) of \(V(G)\), let \(\pi = (v_1, v_2, \ldots, v_n, v_{n+1} = v_1)\), and let \(d(\pi) = \sum\limits_{i=1}^n d(v_i, v_{i+1})\). The set of possible values of \(d(\pi)\) of all cyclic orderings \(\pi\) of \(V(G)\) is called the Hamiltonian spectrum of \(G\). We determine the Hamiltonian spectrum for any tree.

A digraph \(D(V, E)\) is said to be graceful if there exists an injection \(f : V(D) \rightarrow \{0, 1, \ldots, |V|\}\) such that the induced function \(f’ : E(D) \rightarrow \{1, 2, \ldots, |V|\}\) which is defined by \(f'(u,v) = [f(v) – f(u)] \pmod{|E| + 1}\) for every directed edge \((u,v)\) is a bijection. Here, \(f\) is called a graceful labeling (graceful numbering) of digraph \(D(V, E)\), while \(f’\) is called the induced edge’s graceful labeling of digraph \(D(V,E)\). In this paper, we discuss the gracefulness of the digraph \(n-\vec{C}_m\) and prove that the digraph \(n-\vec{C}_{17}\) is graceful for even \(n\).

Candelabra quadruple systems, which are usually denoted by \(\text{CQS}(g^n : s)\), can be used in recursive constructions to build Steiner quadruple systems. In this paper, we introduce some necessary conditions for the existence of a \(\text{CQS}(g^n : s)\) and settle the existence when \(n = 4,5\) and \(g\) is even. Finally, we get that for any \(n \in \{n \geq 3: n \equiv 2,6 \pmod{12}\) and \(n \neq 8\}\), there exists a \(\text{CQS}(g^n : s)\) for all \(g \equiv 0 \pmod{6}\), \(s \equiv 0 \pmod{2}\) and \(0 \leq s \leq g\).

Let \(G\) be a non-abelian group and let \(Z(G)\) be the center of \(G\). Associate with \(G\) a graph \(\Gamma_G\) as follows: Take \(G\setminus Z(G)\) as vertices of \(\Gamma_G\) and join two distinct vertices \(x\) and \(y\) whenever \(xy \neq yx\). Graph \(\Gamma_G\) is called the non-commuting graph of \(G\) and many of graph theoretical properties of \(\Gamma_G\) have been studied. In this paper, we study some metric graph properties of \(\Gamma_G\).

For integers \(p\), \(q\), \(s\) with \(p \geq q \geq 2\) and \(s \geq 0\), let \(\mathcal{K}_{2}^{-s}(p,q)\) denote the set of \(2\)-connected bipartite graphs which can be obtained from the complete bipartite graph \(K_{p,q}\) by deleting a set of \(s\) edges. F.M.Dong et al. (Discrete Math. vol.\(224 (2000) 107-124\)) proved that for any graph \(G \in \mathcal{K}_{2}^{-s}(p,q)\) with \(p \geq q \geq 3\) and \(0 \leq s \leq \min\{4, q-1\}\), then \(G\) is chromatically unique. In \([13]\), we extended this result to \(s = 5\) and \(s = 6\). In this paper, we consider the case when \(s = 7\).

Let \(\lambda K_{h^u}\) denote the \(\lambda\)-fold complete multipartite graph with \(u\) parts of size \(h\). A cube factorization of \(\lambda K_{h^u}\) is a uniform \(3\)-factorization of \(\lambda K_{h^u}\) in which the components of each factor are cubes. We show that there exists a cube factorization of \(\lambda K_{h^u}\) if and only if \(uh \equiv 0 \pmod{8}\), \(\lambda (u-1)h \equiv 0 \pmod{3}\), and \(u \geq 2\). It gives a new family of uniform \(3\)-factorizations of \(\lambda K_{h^u}\). We also establish the necessary and sufficient conditions for the existence of cube frames of \(\lambda K_{h^u}\).

We recall from [13] a shell graph of size \(n\), denoted \(C(m,n-3)\), is the graph obtained from the cycle \(C_n(v_0,v_1,v_2\ldots,v_{n-1})\) by adding \(m-3\) consecutive chords incident at a common vertex, say \(v_0\). The vertex \(v_0\) of \(C(n,n-3)\) is called the apex of the shell \(C(n,n-3)\). The vertex \(v_0\) of \(C(n,n-3)\) is said to be at level \(l\).

A graph \(C(2n,n-2)\) is called an alternate shell, if \(C(2n,n-2)\) is obtained from the cycle \(C{2n}(v_0,v_1,v_2\ldots,v_{2n-1})\) by adding \(n-2\) chords between the vertex \(v_0\) and the vertices \(v_{2i-1}\) for \(1-i\delta n\). If the vertex \(v_i\) of \(C(2n,n-2)\) at level \(l\) and is adjacent with \(v_0\), then \(v_l\) is said to be at level \(l\) with a chord, otherwise the vertex \(v_i\) is said to be at level \(l\) without a chord.

A graph, denoted \(G{2n_i,n_i,2,k,l}\), is called one vertex union of alternate shells with a path at any common level \(l\) (with or without chords), if it is obtained from \(k\) alternate shells \(C(2n_i,n_i-2)’s\), \(1- i\delta k\), by merging them together at their apex and joining \(k\) vertices each chosen from a distinct alternate shell in a particular level \(l\) (with or without chords) by a path \(P_{2k-1}\), such that the chosen vertex of the \(i\)th alternate shell \(C(2n_i,n_i-2)\) is at the \((2i-1)\)th vertex of the \(P_{2k-l}\) for \(1- i\delta k\). We denote the graph \(G{2n_i,n_i,2,k,l}\) as \(G{2n_i,n_i,2,k,l_c}\) if the path \(P_{2k-1}\) joins the vertices only at the common level \(l\) with chords.

In this paper, we show that \(G{2n_i,n_i,2,k,l_c}\) is graceful and admits an \(A\)-labeling, for \(k-\tau1, n_i\), \( 3,1\tau1,n_i\), and \(G{2n_i,n_i,2,k,1}\) is cordial, for \(n_i-n-3 ,k-1,1\tau i\).

A \((k,t)\)-list assignment \(L\) of a graph \(G\) is a mapping which assigns a set of size \(k\) to each vertex \(v\) of \(G\) and \(|\bigcup_{v\in V(G)}L(v)| = t\). A graph \(G\) is \((k, t)\)-choosable if \(G\) has a proper coloring \(f\) such that \(f(v) \in L(v)\) for each \((k, t)\)-list assignment \(L\).

We determine \(t\) in terms of \(k\) and \(n\) that guarantee \((k, t)\)-choosability of any \(n\)-vertex graph and a better bound if such graph does not contain a \((k+1)\)-clique.

For paths \(P_n\), Chartrand, Nebesky and Zhang gave the exact value of \(ac'(P_n)\) for \(n \leq 8\), and showed that \(ac'(P_n) \leq \binom{n-2}{2}+2\) for every positive integer \(n\), where \(ac'(P_n)\) denotes the nearly antipodal chromatic number of \(P_n\). In this paper, we determine the exact values of \(ac'(P_n)\) for all even integers \(n \geq 8\).

A \(2\)-factor of a graph \(G\) is a \(2\)-regular spanning subgraph of \(G\) and a \(2\)-factorization of a graph \(G\) is a \(2\)-factor decomposition of \(G\). A complete solution to the problem of determining the spectrum of \(4\)-cycles in \(2\)-factorizations of the complete bipartite graph is presented.

We study the independence number of the Cartesian product of binary trees and more general bipartite graphs. We give necessary and sufficient conditions on bipartite graphs under which certain upper and lower bounds on the independence number of the product are equal. A basic tool will be an algorithm for finding the independence number of a binary tree.

Multireceiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify the authenticity of the received message. In this paper, we construct two multireceiver authentication codes from symplectic geometry over finite fields. The parameters and the probabilities of deceptions of the codes are also computed.

We give determinant expressions of the zeta function and an \(L\)-function of a semiregular weighted bipartite graph. As an application, we present a decomposition formula for the weighted complexity of a semiregular weighted bipartite graph.

In this paper, we characterize the potentially \((K_5 – C_4)\)-graphic sequences, where \(K_s – C_4\) is the graph obtained from \(K_5\) by removing four edges of a \(4\)-cycle \(C_4\). This characterization implies a theorem due to Lai \([6]\).

A graph is said to be cordial if it has a \(0-1\) labeling that satisfies certain properties. The purpose of this paper is to generalize some known theorems and results of cordial graphs. Specifically, we show that certain combinations of paths, cycles, stars, and null graphs are cordial. Finally, we prove that the torus grids are cordial if and only if its size is not congruent to \(2\) \((mod 4)\).

A graph \(G\) is edge-magic if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1, 2, 3, \ldots, |V(G)| + |E(G)|\}\) such that for any edge \(uv\) of \(G\), \(f(u) + f(uv) + f(v)\) is constant. Moreover, \(G\) is super edge-magic if \(V(G)\) receives \(\{1, 2, \ldots, |V(G)|\}\) smallest labels. In this paper, we propose methods for constructing new (super) edge-magic graphs from some old ones by adding some new pendant edges.

In this study, we consider a generalization of the well-known Fibonacci and Lucas numbers related to combinatorial sums by using finite differences. To write generalized Fibonacci and Lucas sequences in a new direct way, we investigate some new properties of these numbers.

A graph \(G\) is called edge-magic if there exists a bijective function \(\phi: V(G) \cup E(G) \rightarrow \{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that \(\phi(x) + \phi(xy) + f\phi(y) = c(\phi)\) is a constant for every edge \(xy \in E(G)\), called the valence of \(\phi\). A graph \(G\) is said to be super edge-magic if \(\phi(V(G)) = \{1, 2, \ldots, |V(G)|\}\). The super edge-magic deficiency, denoted by \(\mu_s(G)\), is the minimum nonnegative integer \(n\) such that \(G \cup nK_1\) has a super edge-magic labeling, if such integer does not exist we define \(\mu_s(G)\) to be \(+\infty\). In this paper, we study the super edge-magic deficiency of some families of unicyclic graphs.

In \([FP]\) the \(ECO\) methed and Aigner’s theory of Catalan-like numbers are compared, showing that it is often possible to translate a combinatorial situation from one theory into the other by means of a standard change of basis in a suitable vector space. In the present work we emphasize the soundness of such an approach by finding some applications suggested by the above mentioned translation. More precisely, we describe a presumably new bijection between two classes of lattice paths and we give a combinatorial interpretation to an integer sequence not appearing in \([SI]\).

High stopping-distance low-density parity-check \((LDPC)\) product codes with finite geometry \(LDPC\) and Hamming codes as the constituent codes are constructed. These codes have high stopping distance compared to some well-known LDPC codes. As examples, linear \((511, 180, 30)\), \((945, 407, 27)\), \((2263, 1170, 30)\), and \((4095, 2101, 54)\) LDPC codes are designed with stopping distances \(30\), \(27\), \(30\), and \(54\), respectively. Due to their good stopping redundancy, they can be considered as low-complexity codes with very good performance when iterative decoding algorithms are used.

The basis number of a graph \(G\) is defined to be the least positive integer \(d\) such that \(G\) has a \(d\)-fold basis for the cycle space of \(G\).

In this paper, we prove that the basis number of the Cartesian product of different ladders is exactly \(4\). However, if we apply Theorem \(4.1\) of Ali and Marougi \([4]\), which is stated in the introduction as Theorem \(1.1\), we find that the basis number of the circular and Möbius ladders with circular ladders and Möbius ladders is less than or equal to \(5\), and the basis number of ladders with circular ladders and circular ladders with circular ladders is at most \(4\).

It is shown that there are \(\binom{2n-r-1}{n-r}\) noncrossing partitions of an \(n\)-set together with a distinguished block of size \(r\), and \(\binom{n}{k-1}\binom{n-r-1}{k-2}\) of these have \(k\) blocks, generalizing a result of Béna on partitions with one crossing. Furthermore, specializing natural \(q\)-analogues of these formulae with \(q\) equal to certain \(d^{th}\) roots of unity gives the number of such objects having \(d\)-fold rotational symmetry.

In this paper, we introduce the concept of geodesic graph at a vertex of a connected graph and investigate its properties. We determine the bounds for the number of edges of the geodesic graph. We prove that an edge of a graph is a cut edge if and only if it is a cut edge of each of its geodesic graphs. Also, we characterize a bipartite graph as well as a geodetic graph in terms of its geodesic graph.

In this paper, we study the circular choosability recently introduced by Mohar \([5]\) and Zhu \([11]\). In this paper, we show that the circular choosability of planar graphs with girth at least \(\frac{10n+8}{3}\) is at most \(2 + \frac{2}{n}\), which improves the earlier results.

An orientation of a simple graph \(G\) is called an oriented graph. If \(D\) is an oriented graph, \(\delta(D)\) its minimum degree and \(\lambda(D)\) its edge-connectivity, then \(\lambda(D) \leq \delta(D)\). The oriented graph is called maximally edge-connected if \(\lambda(D) = \delta(D)\) and super-edge-connected, if every minimum edge-cut is trivial. If \(D\) is an oriented graph with the property that the underlying graph \(G(D)\) contains no complete subgraph of order \(p+1\), then we say that the clique number \(\omega(D)\) of \(D\) is less or equal \(p\).

In this paper, we present degree sequence conditions for maximally edge-connected and super-edge-connected oriented graphs \(D\) with clique number \(\omega(D) \leq p\) for an integer \(p \geq 2\).

A proper total coloring of a graph \(G\) is called Smarandachely adjacent vertex total coloring of graph if for any two adjacent and distinct vertices \(u\) and \(v\) in \(G\), the set of colors assigned to the vertices and the edges incident to \(u\) doesn’t contain the set of colors assigned to the vertices and the edges incident to \(v\), vice versa. The minimal number of colors required for a Smarandachely adjacent vertex total coloring of graph is called the Smarandachely adjacent vertex total chromatic number of graph. In this paper, we define a kind of \(3\)-regular Multilayer Cycle \(Re(n,m)\) and obtain the Smarandachely adjacent vertex total chromatic number of it.

A perfectly one-factorable (PIF) regular graph \(G\) is a graph admitting a partition of the edge-set into one-factors such that the union of any two of them is a Hamiltonian cycle. We consider the case in which \(G\) is a cubic graph. The existence of a PIF cubic graph is guaranteed for each admissible value of the number of vertices. We give conditions for determining PIF graphs within a subfamily of generalized Petersen graphs.

In this paper, we give the generalization \(\{G_{k,n}\}_{n\in N }\) of \(k\)-Fibonacci and \(k\)-Lucas numbers. After that, by using this generalization, some new algebraic properties on these numbers have been obtained.

Let \(K_q(n, R)\) denote the least cardinality of a \(q\)-ary code of length \(n\), such that every \(q\)-ary word of length \(n\) differs from at least one word in the code in at most \(R\) places. We use a method of Blass and Litsyn to derive the bounds \(K_4(5,2) \geq 14\) and \(K_4(6,2) \geq 32\).

Let \(d_{q}(n,k)\) be the maximum possible minimum Hamming distance of a linear \([n, k]\) code over \(\mathbb{F}_q\). Tables of best known linear codes exist for all fields up to \(q = 9\). In this paper, linear codes over \(\mathbb{F}_{11}\) are constructed for \(k\) up to \(7\). The codes constructed are from the class of quasi-twisted codes. These results show that there exists a \((78,8)\) arc in \(\text{PG}(2,11)\). In addition, the minimum distances of the extended quadratic residue codes of lengths \(76\), \(88\) and \(108\) are determined.