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.

The niche graph of a digraph \(D\) is the undirected graph defined on the same vertex set in which two vertices are adjacent if they share either a common in-neighbor or a common out-neighbor in \(D\). We define a hierarchy of graphs depending on the condition of being the niche graph of a digraph having, respectively, no cycles, no cycles of length two, no loops, or loops. Our goal is to classify in this hierarchy all graphs of order \(n \geq 3\) having a subgraph isomorphic to \(K_{n-2}\).

Let \(\mathcal{H}_1, \ldots, \mathcal{H}_t\) be classes of graphs. The class Ramsey number \(R(\mathcal{H}_1, \ldots, \mathcal{H}_t)\) is the smallest integer \(n\) such that for each \(t\)-edge colouring \((G_1, \ldots, G_t)\) of \(K_n\), there is at least one \(i \in \{1, \ldots, t\}\) such that \(G_i\) contains a subgraph \(H_i \in \mathcal{H}_i\). We take \(t = 2\) and determine \(R(\mathcal{G}^1_l, \mathcal{G}^1_m)\) for all \(2 \leq l \leq m\) and \(R(\mathcal{G}^2_i, \mathcal{G}^2_{m})\) for all \(3 \leq l \leq m\), where \(\mathcal{G}^i_j\) consists of all edge-minimal graphs of order \(j\) and minimum degree \(i\).

Let \(G\) be a \(2\)-connected graph with a toroidal rotation system given. An algorithm for constructing a straight line drawing with no crossings on a rectangular representation of the torus is presented. It is based on Read’s algorithm for constructing a planar layout of a \(2\)-connected graph with a planar rotation system. It is proved that the method always works. The complexity of the algorithm is linear in the number of vertices of \(G\).

A graph \(G\) is called super-edge-magic if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that \(f(u) + f(v) + f(uv) = C\) is a constant for any \(uv \in E(G)\) and \(f(V(G)) = \{1, 2, \ldots, |V(G)|\}\). In this paper, we show that the generalized Petersen graph \(P(n, k)\) is super-edge-magic if \(n \geq 3\) is odd and \(k = 2\).

We reprove an important case of a recent topological result on improved Bonferroni inequalities due to Naiman and Wynn in a purely combinatorial manner. Our statement and proof involves the combinatorial concept of non-evasiveness instead of the topological concept of contractibility. In contradistinction to the proof of Naiman and Wynn, our proof does not require knowledge of simplicial homology theory.