There are 19 connected cubic graphs of order 10. If \(G\) is one of a specific set 3 the 19 graphs. we find necessary and sufficient conditions for the existence of \(G\)-decompositions of \(K_v\).

For a positive integer \(k,\) let \( [k] = {1,2,…,k}\), let \(P([k])\) denote the power set of the set \([k]\) and let \(P*([k]) = P([k]) – {\emptyset}\). For each integer \(t\) with \(1 \le t < k\), let \(P_t([k])\) denote the set of \(t\)-element subsets of \(P([k])\). For an edge coloring \(c : E(G)\to P_t ([k])\) of a graph \(G\), where adjacent edges may be colored the same, \(c' : V(G) \to P*([k])\) is the vertex coloring in which \(c' (v)\) is the union of the color sets of the edges incident with \(v\). If \(c'\) is a proper vertex coloring of \(G\), then \(c\) is a majestic \(t\)-tone k-coloring of \(G\) For a fixed positive integer \(t\), the minimum positive integer \(k\) for which a graph \(G\) has a majestic t-tone k-coloring is the majestic t-tone index \(maj_t (G)\) of \(G\). It is known that if \(G\) is a connected bipartite graph or order at least 3, then \(maj_t(G) = t + 1\) or \(maj_t (G) = t + 2\) for each positive integer t. It is shown that (i) if \(G\) is a 2-connected bipartite graph of arbitrarily large order \(n\) whose longest cycles have length \(l\) where where \(n-5 \leq l \leq n\) and \(t\geq 2\) is an integer, then \(maj_t(G)=t+1\) and (ii) there is a 2-connected bipartite graph F of arbitrarily large order n whose longest cycles have length n-6 and \(maj_2(F)=4\). Furthermore, it is shown for integers \(k,t \ge 2\) that there exists a k-connected bipartite graph \(G\) such that \(maj_t(G) =t+2\). Other results and open questions are also presented.

The 3-path \(P_3(G)\) of a connected graph \(G\) of order 3 or more has the set of all 3-path (path of order 3) of \(G\) as its vertex of \(P_3(G)\) are adjacent if they have a 2-path in common. A Hamiltonian walk in a nontrivial connected graph \(G\) is a closed walk of minimum length that contains every vertex of \(G\). With the aid of spanning trees and Hamiltonian walks in graphs, we provide sufficient conditions for the 3-path graph of a connected graph to be Hamiltonian.

Let \(R\) be a commutative ring with identity. For any integer \(K > 1,\) an element is a \(k\) zero divisor if there are \(K\) distinct elements including the given one, such that the product of all is zero but the product of fewer than all is nonzero. Let \(Z(R,K)\) denote the set of the \(K\) zero divisors of \(R\). A ring with no \(K\)-zero divisors is called a \(K\)-domain. In this paper we define the hyper-graphic constant \(HG(R)\) and study some basic properties of \(K\)-domains. Our main results is theorem 5.1 which is as fellow:

Let \(R\) be a commutative ring such that the total ring of fraction \(T(R)\) is dimensional. If \(R\) is a \(K\)-domain for \(k \geq 2,\) then \(R\) has finitely many minimal prime ideals.

Using the results and lemma 5.4, we improve a finiteness theorem proved in [11] to a more robust theorem 5.5 which says:

Suppose \(R\) is not a \(k\)-domain and has more then \(k\)-minimal prime ideals.

Further, suppose that \(T(R)\) is a zero dimensional ring. Then \(Z(R,K)\) is finite if and only if \(R\) is finite.

We end this paper with a proof of an algorithm describing the maximal \(k\)-zero divisor hypergraphs on \(\mathbb{Z}_n\).

The subject matter for this paper is GDDs with three groups of sizes \(n_1,n(n\geq n_1)\) and \(n+1\), for \(n_1=1\, or\, 2\) and block size four. A block having Configuration \((1,1,2)\) means that the block contains 1 point from two different groups and 2 points from the remaining group. a block having Configuration \((2,2)\) means that the block has exactly two points from two of the three groups. First, we prove that a GDD\((n_1,n,n+1,4;\lambda_1,\lambda_2)\) for \(n_1 = 1\, o\,r 2\) does not exist if we require that exactly halh of the blocks have the Configuration \((1,1,2)\) and the other half of the blocks have the configuration \((2,2)\) except possibly for n=7 when \(n_1=2\). Then we provide necessary conditions for the existence of a GDD\((n_1,n,n+1,4;\lambda_1,\lambda_2)\) for \(n_1=1\, and \,2\), and prove that these conditions are sufficient for several families of GDDs. For \(n_1=2\), these usual necessary conditions are not sufficient in general as we provide a glimpse of challenges which exist even for the case of \(n_1=2\). A general results that a GDD\((n_1,n_2,n_3,4;\lambda_1,\lambda_2)\) exists if \(n_1 + n_2 + n_3=0,4\) \((mod\, 12)\) is also given.

Recently GDDs with two groups and block size four were studied in a paper where the authors constructed two families out of four possible cases with an equal number of even, odd, and group blocks. In this paper, we prove partial existence of one of the two remaining families, namely \(GDD(11t + 1, 2,4; 11t +1, 7t)\), with 7 \(\nmid \)(11t+ 1). In addition, we show a useful construction of \(GDD(6t+ 4, 2, 4; 2, 3)\).

A cancellable number (CN) is a fraction in which a decimal digit can be removed (“‘canceled”) in the numerator and denominator without changing the value of the number; examples include \(\frac{64}{16}\) where the 6 can be canceled and \(\frac{98}{49}\) where the 9 can be canceled. We show that the slope of the line of a cancellable number need not be negative.

Let \( \Gamma \) be a finite group and let \( \Delta \) be a generating set for \( \Gamma \). A Cayley map is an orientable 2-cell imbedding of the Cayley graph \( G_\Delta(\Gamma) \) such that the rotation of arcs emanating from each vertex is determined by a unique cyclic permutation of generators and their inverses. A probability model for the set of all Cayley maps for a fixed group and generating set, where the distribution is uniform. We focus on certain finite abelian groups with generating set chosen as the standard basis. A lower bound is provided for the probability that a Cayley map for such a group and generating set is symmetrical.

A graph is asymmetric if the automorphism group of its set of vertices is trivial. A graph is called non-asymmetric if and only if it is not asymmetric. A graph \( G \) is minimally non-asymmetric if \( G \) is non-asymmetric but \( G – e \) is asymmetric for any edge \( e \) contained in \( G \).

Given a finite set \( V \) (of elements called varieties) and integers \( k \), \( r \), and \( \lambda \), a balanced incomplete block design (BIBD) is a family of \( k \)-element subsets of \( V \), called blocks, such that any element is contained in \( r \) blocks and any pair of distinct varieties \( u \) and \( w \) is contained in exactly \( \lambda \) blocks.

In this paper, we give examples of minimally non-asymmetric graphs constructed from balanced incomplete block designs.

There is a special case of a generalized Clifford algebra, known as a Clifford graph algebra, which is useful for studying a simple graph \( G_n \), with \( n \) vertices. We will discuss how this algebra \( GA(G_n) \) can represent \( G_n \), and prove that it exists in general by defining it as an appropriate sub-algebra of a classical Clifford algebra. We will then refine this process of “construction by inclusion” for the path graph \( P_n \), and the complete star graph \( K_{1,n} \), by choosing from a parent classical Clifford algebra as many bi-vectors as possible for the generators which define \( GA(P_n) \) and \( GA(K_{1,n}) \).