In \(1969\), Dewdney introduced the set \(\Gamma\) of \({primal \;graphs}\), characterized by the following two properties: every finite, simple graph \(G\) is the union of non-isomorphic, edge-disjoint subgraphs of \(G\) so that each of the subgraphs is in \(\Gamma\); and, if \(G\) is in \(\Gamma\), then the only such union consists of \(G\) itself. In the period around 1990, several works concerning the determination of the graphs in \(\Gamma\) were published and one Ph.D. thesis written. However, the classification of the members of \(\Gamma\) remains elusive. The main point of this work is to simplify and unify some of the principal results of Preen’s Ph.D. thesis that generalize earlier results about primal graphs with maximum degree \(2\).

In order to characterize convex polyhedra with regular polygonal faces by a minimal number of parameters, we first introduce some new parameters, then we analyze a table of their values to see how well different sets of parameters tell these solids apart, and finally we present their characterization by four parameters.

In this paper, we show a short proof of the \( q \)-binomial theorem by Schützenberger’s identity with \( q \)-commuting variables.

A non-empty \( r \)-element subset \( A \) of an \( n \)-element set \( X_n \), and a partition \( \pi \) of \( X_n \), are said to be orthogonal if every class of \( \pi \) meets \( A \) in exactly one element. A partition type is determined by the number of classes of each distinct size of the partition. The Johnson graph \( J(n,r) \) is the graph whose vertices are the \( r \)-element subsets of \( X_n \), with two sets being adjacent if they intersect in \( r-1 \) elements. A partition of a given type \( \tau \) is said to be a \( \tau \)-label for an edge \( AB \) in \( J(n,r) \) if the sets \( A \) and \( B \) are orthogonal to the partition. A cycle \( \mathcal{H} \) in the graph \( J(n,r) \) is said to be \( \tau \)-labeled if for every edge of \( \mathcal{H} \), there exists a \( \tau \)-label, and the \( \tau \)-labels associated with distinct edges are distinct. Labeled Hamiltonian cycles are used to produce minimal generating sets for transformation semigroups. We identify a large class of partition types \( \tau \) with a non-zero gap for which every Hamiltonian cycle in the graph \( J(n,r) \) can be \( \tau \)-labeled, showing, for example, that this class includes all the partition types with at least one class of size larger than 3 or at least three classes of size 3.

Let \( G \) be a graph, and \( k \) a positive integer. A graph \( G \) is fractional independent-set-deletable \( k \)-factor-critical (in short, fractional ID-\(k\)-factor-critical) if \( G – I \) has a fractional \( k \)-factor for every independent set \( I \) of \( G \). In this paper, it is proved that if \( \kappa(G) \geq \max \left\{ \frac{k^2 + 6k + 1}{2}, \frac{(k^2 + 6k + 1) \alpha(G)}{4k} \right\} \), then \( G \) is fractional ID-\(k\)-factor-critical.

In this paper, we constructed two multireceiver authentication codes from singular symplectic geometry over finite fields. Under the assumption that the probability distribution on the source states and sender’s key space is uniform, the parameters and success probabilities of different types of deceptions are also computed.

An oriented coloring of a directed graph is a vertex coloring in which no two adjacent vertices belong to the same color class and all of the arcs between any two color classes have the same direction. Injective oriented colorings are oriented colorings that satisfy the extra condition that no two in-neighbors of a vertex receive the same color. The oriented chromatic number of an unoriented graph is the maximum oriented chromatic number over all possible orientations. Similarly, the injective oriented chromatic number of an unoriented graph is the maximum injective oriented chromatic number over all possible orientations. The main results obtained in this paper are bounds on the injective oriented chromatic number of two-dimensional grid graphs.

Let \(\lambda K_{v}\) be the complete multigraph of order \(v\) and index \(\lambda\), where any two distinct vertices \(x\) and \(y\) are joined exactly by \(\lambda\) edges \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-design of \(\lambda K_{v}\), denoted by \((v, G, \lambda)\)-GD, is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\) and \(\mathcal{B}\) is a collection of subgraphs of \(K_{v}\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_{v}\) are joined in exactly \(\lambda\) blocks of \(\mathcal{B}\). There are four graphs with seven vertices, seven edges, and one 5-cycle, denoted by \(G_i\), \(i=1,2,3,4\). In [9], we have solved the existence problems of \((v, G_i, 1)\)-GD. In this paper, we obtain the existence spectrum of \((v, G_i, \lambda)\)-GD for any index \(\lambda\).

The values of the Ramsey numbers \( R(C_4, H) \), for any graph \( H \) on 6 vertices, are shown in [3]. An erratum is corrected in [4,6], giving \( R(C_4, K_{3,3}) = 11 \).

In this paper, we correct three other errata of [3], proving that \( R(C_4, K_1 + (K_{2,3} – e)) = 9 \), \( R(C_4, \overline{K_3 \cup P_3}) = 11 \), and \( R(C_4, \overline{2P_3}) = 11 \), instead of 10.

We use dynamic programming to compute the domination number of the Cartesian product of two directed paths, \( \overrightarrow{P}_m \) and \( \overrightarrow{P}_n \), for \( m \leq 25 \) and all \( n \). This suggests that the domination number for \(\min(m,n) \geq 4\) is \( \left\lfloor \frac{(m+1)(n+1)}{3} \right\rfloor – 1 \), which we then confirm by showing that this is both an upper and a lower bound on the domination number.