Let \(G\) be a \(2\)-connected simple graph with order \(n\) (\(n \geq 5\)) and minimum degree 6. This paper proves that if \(|N(u) \cup N(v)| \geq n – \delta + 2\) for any two nonadjacent vertices \(u, v \in V(G)\), then \(G\) is edge-pancyclic, with a few exceptions. Under the same condition, we prove that if \(u, v \in V(G)\) and \(\{u, v\}\) is not a cut set and \(N(u) \cap N(v) \neq \phi\) when \(uv \in E(G)\), then there exist \(u\)–\(v\) paths of length from \(d(u, v)\) to \(n – 1\).

The purpose of this paper is to extend the well-known concepts of additive permutations and bases of additive permutations to the case when repeated elements are permitted; that means that the basis (an ordered set) can become an ordered multiset. Certain special cases are studied in detail and all bases with repeated elements up to cardinality six are enumerated, together with their additive permutations.

We show how lattice paths and the reflection principle can be used to give easy proofs of unimodality results. In particular, we give a “one-line” combinatorial proof of the unimodality of the binomial coefficients. Other examples include products of binomial coefficients, polynomials related to the Legendre polynomials, and a result connected to a conjecture of Simion.

The search for homometric structures, i.e., non-congruent structures sharing the same autocorrelation function, is shown to be of a combinatorial nature and can be studied using purely algebraic techniques. Several results on the existence of certain homometric structures which contradict a theorem by S. Piccard are proved based on a polynomial representation model and the factorization of polynomials over the rationals. Combinatorial arguments show that certain factorizations do not lead to counterexamples to S. Piccard’s theorem.

Let \(G = (V, E)\) be a graph. For any real valued function \(f: V \to \mathbb{R}\) and \(S \subseteq V\), let \(f(S) = \sum_{u \in S} f(u)\). Let \(c, d\) be positive integers such that \(\gcd(c, d) = 1\) and \(0 < \frac{c}{d} \leq 1\). A \(\frac{c}{d}\)-dominating function \(f\) is a function \(f: V \to \{-1, 1\}\) such that \(f[v] \geq 1\) for at least \(\frac{c}{d}\) of the vertices \(v \in V\). The \(\frac{c}{d}\)-domination number of \(G\), denoted by \(\gamma_{\frac{c}{d}}(G)\), is defined as \(\min\{f(V) | f\) is a \(\frac{c}{d}\)-dominating function on \(G\}\). We determine a sharp lower bound on \(\gamma_{\frac{c}{d}}(G)\) for regular graphs \(G\), determine the value of \(\gamma_{\frac{c}{d}}(G)\) for an arbitrary cycle \(C_n\), and show that the decision problem PARTIAL SIGNED DOMINATING FUNCTION is \(NP\)-complete.

The vertex set of a halved cube \(Q’_d\) consists of a bipartition vertex set of a cube \(Q_d\) and two vertices are adjacent if they have a common neighbour in the cube. Let \(d \geq 5\). Then it is proved that \(Q’_d\) is the only connected, \(\binom{d}{3}\)-regular graph on \(2^d\) vertices in which every edge lies in two \(d\)-cliques and two \(d\)-cliques do not intersect in a vertex.

Geometrical representations of certain classical number tables modulo a given prime power (binomials, Gaussian \(g\)-binomials and Stirling numbers of \(1st\) and \(2nd\) kind) generate patterns with self-similarity features. Moreover, these patterns appear to be strongly related for all number tables under consideration, when a prime power is fixed.

These experimental observations are made precise by interpreting the recursively defined number tables as the output of certain cellular automata \((CA)\). For a broad class of \(CA\) it has been proven \([11]\) that the long time evolution can generate fractal sets, whose properties can be understood by means of hierarchical iterated function systems. We use these results to show that the mentioned number tables (mod \(p^v\)) induce fractal sets which are homeomorphic to a universal fractal set denoted by \(\mathcal{S}_{p^v}\) which we call Sierpinski triangle (mod \(p^v\)).