A strong \( k \)-edge-coloring of a graph \( G \) is an assignment of \( k \) colors to the edges of \( G \) in such a way that any two edges meeting at a common vertex, or being adjacent to the same edge of \( G \), are assigned different colors. The strong chromatic index of \( G \) is the smallest number \( k \) for which \( G \) has a strong \( k \)-edge-coloring. A Halin graph is a planar graph consisting of a tree with no vertex of degree two and a cycle connecting the leaves of the tree. A caterpillar is a tree such that the removal of the leaves becomes a path. In this paper, we show that the strong chromatic index of cubic Halin graph is at most 9. That is, every cubic Halin graph is edge-decomposable into at most 9 induced matchings. Also, we study the strong chromatic index of a cubic Halin graph whose characteristic tree is a caterpillar.

Let \( G \) be a graph of order \( n(G) \), minimum degree \( \delta(G) \), diameter \( d_m(G) \), and let \( \bar{G} \) be the complement of the graph \( G \). A vertex set \( D \) is called a dominating set of \( G \), if each vertex not in \( D \) has at least one neighbor in \( D \). The domination number \( \gamma(G) \) equals the minimum cardinality of a dominating set of \( G \).
In this article we show the inequalities

1. \( \gamma(G) \leq \left\lfloor \frac{n(G)}{3} \right\rfloor, \text{ if } \delta(G) \geq 7, \)
2. \( \gamma(G) + \gamma(\bar{G}) \leq \left\lfloor \frac{n(G)}{3} \right\rfloor + 2, \text{ if } \delta(G), \delta(\bar{G}) \geq 7, \text{ and} \)
3. \( \gamma(G) \leq \left\lceil \frac{n(G)}{4} \right\rceil + 1, \text{ if } \text{dm}(G) = 2. \)

Using the concept of connectivity, we present some related upper bounds for the domination number of graphs with \( \text{dm}(G) = 2 \) and \( \text{dm}(G) = 3 \).

We prove in this note that certain caterpillars with diameter 4 or 5 do not factorize complete graphs. This together with results by Kovarova [2,3] and Kubesa [5] gives the complete characterization of the caterpillars with diameter 4 that factorize the complete graph \( K_{2n} \). For diameter 5, we again complement results by Kovarova [4] and Kubesa [6-9] to give the complete characterization for certain class of caterpillars.

High-performance computers have been in great demand for applications in different areas. The increase in the processing power of processors cannot solely satisfy our demand. Parallel computers are made to overcome this technology limitation. In the last decade, research topics on parallel computer using network-connected multicomputer have been studied extensively. A cost-efficient high-speed multicomputer system was built using the SCSI bus for the network connection, and it has been shown that it can reduce the communication overheads and hence increase the overall performance [5]. In order to build highly scalable multiple computers based on this design, we have to take into consideration of different network topologies. Since SCSI bus [2,3] possesses some unique properties, it induces some interesting properties on the design of the network topology. In this paper, we evaluate the performance of the large scale SCSI networks with linear and mesh structures.

The degree set of a finite simple graph \( G \) is the set of distinct degrees of vertices of \( G \). For any given finite set \( \mathcal{D} \) of positive integers, we determine all positive integers \( n \) such that \( \mathcal{D} \) is the degree set of some simple graph with \( n \) vertices. This extends a theorem of Kapoor, Polimeni \& Wall (1977) which shows that the least such \( n \) is \( 1 + \max(\mathcal{D}) \).

Every labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge \( (x,y) \) is the absolute value of the difference of the labels of \( x \) and \( y \). By analogy with graceful labelings, we say that a labeling of the vertices of a graph of order \( n \) is minimally \( k \)-equitable if the vertices are labelled with \( 1, 2, \ldots, n \) and in the induced labeling of its edges every label either occurs exactly \( k \) times or does not occur at all. For \( m \geq 3 \), let \( C_m’ \) (denoted also in the literature by \( C_m \circ K_1 \) and called a corona graph) be a graph with \( 2m \) vertices such that there is a partition of them into sets \( U \) and \( V \) of cardinality \( m \), with the property that \( U \) spans a cycle, \( V \) is independent and the edges joining \( U \) to \( V \) form a matching. Let \( \mathcal{P} \) be the set of all pairs \( (m, k) \) of positive integers such that \( k \) is a proper divisor of \( 2m \) (i.e., a divisor different from \( 2m \) and \( 1 \)) and \( k \) is odd if \( m \) is odd. We show that \( C_m’ \) is minimally \( k \)-equitable if and only if \( (m,k) \in \mathcal{P} \).

We show that the number of points at distance \( i \) from a given point \( x \) in a dense near polygon only depends on \( i \) and not on the point \( x \). We give a number of easy corollaries of this result. Subsequently, we look to the case of dense near polygons \( S \) with an order in which there are two possibilities for \( t_Q \), where \( Q \) is a quad of \( S \), and three possibilities for \( (t_H, v_H) \), where \( H \) is a hex of \( S \). Using the above-mentioned results, we will show that the number of quads of each type through a point is constant. We will also show that the number of hexes of each type through a point is constant if a certain matrix is nonsingular. If each hex is a regular near hexagon, a glued near hexagon or a product near hexagon, then that matrix turns out to be nonsingular in all but one of the eight possible cases. For the exceptional case, however, we provide an example of a near polygon that does not have a constant number of hexes of each type through each point.

In the Euclidean plane, let \( A \), \( B \), \( C \) be noncollinear points and \( T \) be the union of the lines \( AB \), \( BC \), \( CA \). It is shown that there is a point \( P \) such that if \( \tilde{T} \) is the image of \( T \) by any nonrotating uniform expansion about \( P \), then \( T \cap \tilde{T} \) is generally a six-point set that lies on a circle.

We show that for each positive integer \( t \), for which there is a skew-type Hadamard matrix of order \( 4t \), there is a quasi-symmetric \( ((4t – 1)^2, (4t – 1)(2t – 1), t(4t – 3)) \) design.

The Moore upper bound for the order \( n(\Delta, 2) \) of graphs with maximum degree \( \Delta \) and diameter two is \( n(\Delta, 2) < \Delta^2 + 1 \). The only general lower bound for vertex symmetric graphs is \( n_{vt}(\Delta, 2) \geq \left\lfloor \frac{\Delta + 2}{2} \right\rfloor \left\lceil \frac{\Delta + 2}{2} \right\rceil \). Recently, a construction of vertex transitive graphs of diameter two, based on voltage graphs, with order \( \frac{8}{9} \left( \Delta + \frac{1}{2} \right)^2 \) has been given in [5] for \( \Delta = \frac{3q – 1}{2} \) and \( q \) a prime power congruent with 1 mod 4. We give an alternative geometric construction which provides vertex transitive graphs with the same parameters and, when \( q \) is a prime power not congruent to 1 modulo 4, it gives vertex transitive graphs of diameter two and order \( \frac{1}{2} (\Delta + 1)^2 \), where \( \Delta = 2q – 1 \). For \( q = 4 \), we obtain a vertex transitive graph of degree 6 and order 32.