We propose an original approach to the problem of rank-unimodality for Dyck lattices. It is based on a well-known recursive construction of Dyck paths originally developed in the context of the ECO methodology, which provides a partition of Dyck lattices into saturated chains. Even if we are not able to prove that Dyck lattices are rank-unimodal, we describe a family of polynomials (which constitutes a polynomial analog of ballot numbers) and a succession rule which appear to be useful in addressing such a problem. At the end of the paper, we also propose and begin a systematic investigation of the problem of unimodality of succession rules.

A Roman dominating function on a graph \( G \) is a labeling \( f: V(G) \to \{0, 1, 2\} \) such that every vertex with label \( 0 \) has a neighbor with label \( 2 \). The weight of a Roman dominating function is the value \( f(V(G)) = \sum_{u \in V(G)} f(u) \). The minimum weight of a Roman dominating function on a graph \( G \) is called the Roman domination number, denoted by \( \gamma_R(G) \). The Roman bondage number of a graph \( G \) is the cardinality of a smallest set of edges whose removal results in a graph with Roman domination number greater than that of \( G \).

In this paper, we initiate the study of the Roman fractional bondage number, and we present different bounds on Roman fractional bondage. In addition, we determine the Roman fractional bondage number of some classes of graphs.

We show that the principal results of the article “The metric dimension of graphs with pendant edges” [Journal of Combinatorial Mathematics and Combinatorial Computing, 65 (2008) 139-145] do not hold. In this paper, we correct the results and we solve two open problems described in the above-mentioned paper.

Using the definition of the representation number of a graph modulo integers given by Erdős and Evans, we establish the representation number of a complete graph minus a set of disjoint stars. The representation number of a graph \( G \) is the smallest positive integer \( n \) for which there is a labeling of every vertex of \( G \) with a distinct element of \( \{0,1,2,\ldots,n-1\} \) such that two vertices are adjacent if and only if the difference of their labels is relatively prime to \( n \). We apply known results to a complete graph minus a set of stars to establish a lower bound for the representation number; then show a systematic labeling of the vertices producing a representation that attains that lower bound. Thus showing that for complete graphs minus a set of disjoint stars, the established lower bound of the representation number modulo \( n \) is indeed the representation number of the graph. Since the representation modulo an integer for a complete graph minus disjoint stars is attained using the fewest number of primes allowed by the lower bound, it follows that the corresponding Prague dimension will be determined by the largest star removed from the complete graph.

Let \(\lambda K_v\) be the complete multigraph of order \(v\) and index \(\lambda\), where any two distinct vertices \(x\) and \(y\) are joined exactly by \(\lambda\) edges \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-design of \(\lambda K_v\), denoted by \((v,G,\lambda)\)-GD, is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(\lambda K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in exactly \(\lambda\) blocks of \(\mathcal{B}\). There are four graphs which are a 6-circle with two pendant edges, denoted by \(G_i\), \(i = 1,2,3,4\). In [9], we have solved the existence problems of \((v, G_i, 1)\)-GD. In this paper, we obtain the existence spectrum of \((v, G_i, \lambda)\)-GD for any \(\lambda > 1\).

The decycling index of a digraph is the minimum number of arcs whose removal yields an acyclic digraph. The maximum arc decycling number \(\overline{\nabla}'(m,n)\) is the maximum decycling index among all \(m\times n\) bipartite tournaments. Recently, R.C. Vandell determined the numbers \(\overline{\nabla}'(2,n)\), \(\overline{\nabla}'(3,n)\), and \(\overline{\nabla}'(4,n)\) for all positive integers \(n\), as well as \(\overline{\nabla}'(5,5)\). In this work, we use a computer program to obtain \(\overline{\nabla}'(5,6)\), \(\overline{\nabla}'(6,6)\), and \(\overline{\nabla}'(5,7)\), as well as some results on \(\overline{\nabla}'(6,7)\) and \(\overline{\nabla}'(5,8)\). In particular, \(\overline{\nabla}'(6,6) = 10\), and this confirms a conjecture of Vandell.

Let \( G = (V, E) \) be a graph. A function \( f: V \to \{-1, 1\} \) is called a signed dominating function on \( G \) if \( \sum_{u \in N_G[v]} f(u) \geq 1 \) for each \( v \in V \), where \( N_G[v] \) is the closed neighborhood of \( v \). A set \( \{f_1, f_2, \ldots, f_d\} \) of signed dominating functions on \( G \) is called a signed dominating family (of functions) on \( G \) if \( \sum_{i=1}^d f_i(v) \leq 1 \) for each \( v \in V \). The signed domatic number of \( G \) is the maximum number of functions in a signed dominating family on \( G \). The signed total domatic number is defined similarly, by replacing the closed neighborhood \( N_G[v] \) with the open neighborhood \( N_G(v) \) in the definition. In this paper, we prove that the problems of computing the signed domatic number and the signed total domatic number of a given graph are both NP-hard, even if the graph has bounded maximum degree. To the best of our knowledge, these are the first NP-hardness results for these two variants of the domatic number.