Assume that \(G = (V, E)\) is an undirected and connected graph, and consider \(C \subseteq V\). For every \(v \in V\), let \(I_r(v) = \{u \in C: d(u,v) \leq r\}\), where \(d(u,v)\) denotes the number of edges on any shortest path between \(u\) to \(v\) in \(G\). If all the sets \(I_r(v)\) for \(v \in V\) are pairwise different, and none of them is the empty set, \(C\) is called an \(r\)-identifying code. In this paper, we consider \(t\)-vertex-robust \(r\)-identifying codes of level \(s\), that is, \(r\)-identifying codes such that they cover every vertex at least \(s\) times and the code is vertex-robust in the sense that \(|I_r(u) \Delta I_r(v)| \geq 2t+1\) for any two different vertices \(u\) and \(v\). Vertex-robust identifying codes of different levels are examined, in particular, of level \(3\). We give bounds (sometimes exact values) on the density or cardinality of the codes in binary hypercubes and in some infinite grids.

A clique \(C\) is an extreme clique of an interval graph \(G\) if there exists some interval model of \(G\) in which \(C\) is the first clique. A graph \(G\) is homogeneously clique-representable if all cliques of \(G\) are extreme cliques. In this paper, we present characterizations of extreme cliques and homogeneously clique-representable graphs.

In this note, we show that there is no \((945, 177, 33)\)-difference set in any group \(G\) of order \(945\) with a normal subgroup \(K\) such that \(G/K \cong \mathbb{C}_{27} \times \mathbb{C}_5\), and hence no cyclic difference set with such parameters exists. This fills one entry of Baumert and Gordon’s table with “No”.

The study of patterns in permutations is a very active area of current research. Klazar defined and studied an analogous notion of pattern for set partitions. We continue this work, finding exact formulas for the number of set partitions which avoid certain specific patterns. In particular, we enumerate and characterize those partitions avoiding any partition of a 3-element set. This allows us to conclude that the corresponding sequences are P-recursive. Finally, we define a second notion of pattern in a set partition, based on its restricted growth function. Related results are obtained for this new definition.

Let \(G = (V(G), E(G))\) be a graph with \(\delta(G) \geq 1\). A set \(D \subseteq V(G)\) is a paired-dominating set if \(D\) is a dominating set and the induced subgraph \(G[D]\) contains a perfect matching. The paired domination number of \(G\), denoted by \(\gamma_p(G)\), is the minimum cardinality of a paired-dominating set of \(G\). The paired bondage number, denoted by \(b_p(G)\), is the minimum cardinality among all sets of edges \(E’ \subseteq E\) such that \(\delta(G – E’) \geq 1\) and \(\gamma_p(G – E’) > \gamma_p(G)\). For any \(b_p(G)\) edges \(E’ \subseteq E\) with \(\delta(G – E’) \geq 1\), if \(\gamma_p(G – E’) > \gamma_p(G)\), then \(G\) is called uniformly pair-bonded graph. In this paper, we prove that there exists uniformly pair-bonded tree \(T\) with \(b_p(T) = k\) for any positive integer \(k\). Furthermore, we give a constructive characterization of uniformly pair-bonded trees.

A new construction of a B-T unital using Hermitian curves and certain hypersurfaces of \(\text{PG}(3,q^2)\) is presented. Some properties of an algebraic curve containing all points of a B-T unital are also examined.

A construction of optimal quaternary codes from symmetrical Balanced Incomplete Block (BIB) design \((4t – 1, 2t – 1, t – 1)\) is described.

For integers \(s,t \geq 1\), the Ramsey number \(R(s, t)\) is defined to be the least positive integer \(n\) such that every graph on \(n\) vertices contains either a clique of order \(s\) or an independent set of order \(t\). In this note, we derive new lower bounds for the Ramsey numbers: \(R(6,8) \geq 129\), \(R(7,9) \geq 235\) and \(R(8,17) \geq 937\). The new bounds are obtained with a constructive method proposed by Xu and Xie et al. and the help of computer algorithm.

We pursue the problem of counting the imbeddings of a graph in each of the orientable surfaces. We demonstrate how to achieve this for an iterated amalgamation of arbitrarily many copies of any graph whose genus distribution is known and further analyzed into a partitioned genus distribution. We introduce the concept of recombinant strands of face-boundary walks, and we develop the use of multiple production rules for deriving simultaneous recurrences. These two ideas are central to a broad-based approach to calculating genus distributions for graphs synthesized from smaller graphs.

The super (resp., edge-) connectivity of a connected graph is the minimum cardinality of a vertex-cut (resp., an edge-cut) whose removal does not isolate a vertex. In this paper, we consider the two parameters for a special class of graphs \(G(G_p,G_1; M)\), proposed by Chen et al [Applied Math. and Computation, \(140 (2003), 245-254]\), obtained from two \(k\)-regular \(k\)-connected graphs \(G_p\) and \(G_1\), with the same order by adding a perfect matching between their vertices. Our results improve ones of Chen et al. As applications, the super connectivity and the super edge-connectivity of the \(n\)-dimensional hypercube, twisted cube, cross cube, Möbius cube and locally twisted cube are all \(2n – 2\).