The star graph \(S_n\) and the alternating group graph \(A_n\) are two popular interconnection graph topologies. \(A_n\) has a higher connectivity while \(S_n\) has a lower degree, and the choice between the two graphs depends on the specific requirement of an application. The degree of \(S_n\) can be even or odd, but the degree of \(A_n\) is always even. We present a new interconnection graph topology, split-star graph \(S^2_{n}\), whose degree is always odd. \(S^2_{n}\) contains two copies of \(A_n\) and can be viewed as a companion graph for \(A_n\). We demonstrate that this graph satisfies all the basic properties required for a good interconnection graph topology. In this paper, we also evaluate \(S_n\), \(A_n\), and \(S^2_{n}\) with respect to the notion of super connectivity and super edge-connectivity.

We construct a small table of lower bounds for the maximum number of mutually orthogonal frequency squares of types \(F(n; \lambda)\) with \(n \leq 100\).

A graph \(G\) is \(\{R, S\}\)-free if \(G\) contains no induced subgraphs isomorphic to \(R\) or \(S\). The graph \(Z_1\) is a triangle with a path of length \(1\) off one vertex; the graph \(Z_2\) is a triangle with a path of length \(2\) off one vertex. A graph that is \(\{K_{1,3}, Z_1\}\)-free is known to be either a cycle or a complete graph minus a matching. In this paper, we investigate the structure of \(\{K_{1,3}, Z_2\}\)-free graphs. In particular, we characterize \(\{K_{1,3}, Z_2\}\)-free graphs of connectivity \(1\) and connectivity \(2\).

The problem is to determine the number of `cops’ needed to capture a `robber’ where the game is played with perfect information with the cops and the robber alternating moves. The `cops’ capture the `robber’ if one of them occupies the same vertex as the robber at any time in the game. Here we show that a graph with strong isometric dimension two requires no more than two cops.

Combinatorial properties of the multi-peg Tower of Hanoi problem on \(n\) discs and \(p\) pegs are studied. Top-maps are introduced as maps which reflect topmost discs of regular states. We study these maps from several points of view. We also count the number of edges
in graphs of the multi-peg Tower of Hanoi problem and in this way obtain some combinatorial identities.

A given nonincreasing sequence \(\mathcal D = (d_1, d_2, \dots, d_n)\) is said to contain a (nonincreasing) repetition sequence \(\mathcal D ^* = (d_{i_1},d_{i_2} \dots, d_{i_k})\) for some \(k \leq n – 2\) if all values of \(\mathcal D – \mathcal D ^*\) are distinct and for any \(d_{i_i} \in \mathcal D ^*\), there exists some \(d_t \in \mathcal D – \mathcal D ^*\) such that \(d_{i_1} = d_t\). For any pair of integers \(n\) and \(k\) with \(n \geq k + 2\), we investigate the existence of a graphic sequence which contains a given repetition sequence. Our main theorem contains the known results for the special case \(d_{i_1} = d_{i_k}\) if \(k = 1\) or \(k = 2\) (see [1, 5, 2]).

It is shown that the necessary conditions are sufficient for the existence of \(c\)-BRD(\(v, 3, \lambda\)) for all \(c \geq -1\). This was previously known for \(c = 0\) and for \(c = 1\).

Let \(\mathcal{S}\) be the set of vectors \(\{{e^{i\theta}}:\theta=0, \frac{n}{3}, \frac{2n}{3}\}\), and let \(\mathcal{S}\) be a nonempty simply connected union of finitely many convex polygons whose edges are parallel to vectors in \(\mathcal{S}\). If every three points of \(\mathcal{S}\) see a common point via paths which are permissible (relative to \(\mathcal{S}\)), then \(\mathcal{S}\) is star-shaped via permissible paths. The number three is best possible.