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.

Let \(G\) be a graph with \(n\) vertices and suppose that for each vertex \(v\) in \(G\), there exists a list of \(k\) colors, \(L(v)\), such that there is a unique proper coloring for \(G\) from this collection of lists, then \(G\) is called a uniquely \(k\)-list colorable graph. Recently, M. Mahdian and E.S. Mahmoodian characterized uniquely \(2\)-list colorable graphs. Here, we state some results which will pave the way in characterization of uniquely \(k\)-list colorable graphs. There is a relationship between this concept and defining sets in graph colorings and critical sets in latin squares.

Let \(d_3(n,k)\) be the maximum possible minimum Hamming distance of a ternary linear \([n, k, d; 3]\) code for given values of \(n\) and \(k\). The nonexistence of \([142, 7, 92; 3]\), \([162, 7, 106; 3]\), \([165, 7, 108; 3]\), and \([191, 7, 125; 3]\) codes is proved.