By applying the method of generating function, the purpose of this paper is to give several summations of reciprocals related to \(l-th\) power of generalized Fibonacci sequences. As applications, some identities involving Fibonacci, Lucas numbers are obtained.

Bricks are polyominoes with labelled cells. The problem whether a given set of bricks is a code is undecidable in general. We consider sets consisting of square bricks only. We have shown that in this setting, the codicity of small sets (two bricks) is decidable, but \(15\) bricks are enough to make the problem undecidable. Thus the step from words to even simple shapes changes the algorithmic properties significantly (codicity is easily decidable for words). In the present paper we are interested whether this is reflected by quantitative properties of words and bricks. We use their combinatorial properties to show that the proportion of codes among all sets is asymptotically equal to \(1\) in both cases.

Let \(G_{n,m} = C_n \times P_m\), be the cartesian product of an \(n\)-cycle \(C_n\) and a path \(P_m\) of length \(m-1\). We prove that \(\chi'(G_{n,m}) = \chi'(G_{n,m}) = 4\) if \(m \geq 3\), which implies that the list-edge-coloring conjecture (LLECC) holds for all graphs \(G_{n,m}\).

Various authors have defined statistics on Dyck paths that lead to generalizations of the Catalan numbers. Three such statistics are area, maj, and bounce. Haglund, whe introduced the bounce statistic, gave an algebraic proof that \(n(n – 1)/2+\) area — bounce and maj have the same distribution on Dyck paths of order \(n\). We give an explicit bijective proof of the same result.

We develop a new type of a vertex labeling of graphs, namely \(2n\)-cyclic blended labeling, which is a generalization of some previously known labelings. We prove that a graph with this labeling factorizes the complete graph on \(2nk\) vertices, where \(k\) is odd and \(n, k > 1\).

Let \(D = (V, E)\) be a primitive digraph. The exponent of \(D\) at a vertex \(u \in V\), denoted by \(\text{exp}_D(u)\), is defined to be the least integer \(k\) such that there is a walk of length \(k\) from \(u\) to \(v\) for each \(v \in V\). Let \(V = \{v_1,v_2,\ldots ,v_n\}\). The vertices of \(V\) can be ordered so that \(\text{exp}_D(v_{i_1}) \leq \text{exp}_D(v_{i_2}) \leq \ldots \leq \text{exp}_D(v_{i_n})\). The number \(\text{exp}_D(v_{i_k})\) is called \(k\)-exponent of \(D\), denoted by \(\text{exp}_D(k)\). In this paper, we completely characterize \(1\)-exponent set of primitive, minimally strong digraphs with \(n\) vertices.

In \([4]\) H. Galana-Sanchez introduced the concept of kernels by monochromatic paths which generalize the concept of kernels. In \([6]\) they proved the necessary and sufficient conditions for the existence of kernels by monochromatic paths of the duplication of a subset of vertices of a digraph, where a digraph is without monochromatic directed circuits. In this paper we study independent by monochromatic paths sets and kernels by monochromatic paths of the duplication. We generalize result from \([6]\) for an arbitrary edge coloured digraph.

Let \(D = (V, E)\) be a primitive, minimally strong digraph. In \(1982\), J. A. Ross studied the exponent of \(D\) and obtained that \(\exp(D) \leq n + s(n – 8)\), where \(s\) is the length of a shortest circuit in \(D\) \([D]\). In this paper, the \(k\)-exponent of \(D\) is studied. Our principle result is that
\[
\exp_D(k) \leq \begin{cases}
k + 1 + s(n – 3), & \text{if } 1 \leq k \leq s, \\\
k + s(n-3), & \text{if } s+1 \leq k \leq n,\\
\end{cases} \\.
\]
with equality if and only if \(D\) isomorphic to the diagraph \(D_{s,n}\) with vertex set \(V(D_{s,n})=\{v_1,v_2,\ldots,v_n\}\) and arc set \(E(D_{s,n})=\{(v_i,v_{i+1}):1\leq i\leq n-1\}\cap \{(v_s,v_1),(v_n,v_2)\}\). If \((s,n-1)\neq 1\),then
\[
\exp_D(k)< \begin{cases} k + 1 + s(n – 3), & \text{if } 1 \leq k \leq s, \\\ k + s(n-3), & \text{if } s+1 \leq k \leq n, \end{cases} \\ \] and if \((s,n-1)=1\), then \(D_{s, n}\) is a primitive, minimally strong digraph on \(n\) vertices with the \(k\)-exponent \[ \exp_D(k)= \begin{cases} k + 1 + s(n – 3), & \text{if } 1 \leq k \leq s, \\\ k + s(n-3), & \text{if } s+1 \leq k \leq n, \end{cases} \\ \] Moreover, we provide a new proof of Theorem \(1\) in \([6]\) and Theorem \(2\) in \([14]\) by applying this result.

Given a finite projective plane of order \(n\). A quadrangle consists of four points \(A, B, C, D\), no three collinear. If the diagonal points are non-collinear, the quadrangle is called a non-Fano quad. A general sum of squares theorem is proved for the distribution of points in a plane containing a non-Fano quad, whenever \(n \geq 7\). The theorem implies that the number of possible distributions of points in a plane of order \(n\) is bounded for all \(n \geq 7\). This is used to give a simple combinatorial proof of the uniqueness of \(PP(7)\).

Let \(G = (V, E)\) be a graph with \(n\) vertices. The clique graph of \(G\) is the intersection graph \(K(G)\) of the set of all (maximal) cliques of \(G\) and \(K\) is called the clique operator. The iterated clique graphs \(K^*(G)\) are recursively defined by \(K^0(G) = G\) and \(K^i(G) = K(K^{i-1}(G))\), \(i > 0\). A graph is \(K\)-divergent if the sequence \(|V(K^i(G))|\) of all vertex numbers of its iterated clique graphs is unbounded, otherwise it is \(K\)-convergent. The long-run behaviour of \(G\), when we repeatedly apply the clique operator, is called the \(K\)-behaviour of \(G\).

In this paper, we characterize the \(K\)-behaviour of the class of graphs called \(p\)-trees, that has been extensively studied by Babel. Among many other properties, a \(p\)-tree contains exactly \(n – 3\) induced \(4\)-cycles. In this way, we extend some previous results about the \(K\)-behaviour of cographs, i.e., graphs with no induced \(P_4\)s. This characterization leads to a polynomial-time algorithm for deciding the \(K\)-convergence or \(K\)-divergence of any graph in the class.