A \( k \)-edge labeling of a graph \( G \) is a function \( f \) from the edge set \( E(G) \) to the set of integers \( \{0, \ldots, k-1\} \). Such a labeling induces a labeling \( f \) on the vertex set \( V(G) \) by defining \( f(v) = \sum f(e) \), where the summation is taken over all the edges incident on the vertex \( v \) and the value is reduced modulo \( k \). Cahit calls this labeling edge-\( k \)-equitable if \( f \) assigns the labels \( \{0, \ldots, k-1\} \) equitably to the vertices as well as edges.

If \( G_1, \ldots, G_T \) is a family of graphs having a graph \( H \) as an induced subgraph, then by \( H \)-union \( G \) of this family we mean the graph obtained by identifying all the corresponding vertices as well as edges of the copies of \( H \) in \( G_1, \ldots, G_T \).

In this paper we prove that the \( \overline{K}_n \)-union of gears is edge-\( 3 \)-equitable.

Let \( k \) be a positive integer and let \( G \) be a simple graph with vertex set \( V(G) \). If \( v \) is a vertex of \( G \), then the open \( k \)-neighborhood of \( v \), denoted by \( N_{k,G}(v) \), is the set \( N_{k,G}(v) = \{u \mid u \neq v \text{ and } d(u, v) \leq k\} \). The closed \( k \)-neighborhood of \( v \), denoted by \( N_{k,G}[v] \), is \( N_{k,G}[v] = N_{k,G}(v) \cup \{v\} \). A function \( f: V(G) \to \{-1,1\} \) is called a \({signed\; distance \; k -dominating\; function}\) if \( \sum_{u \in N_{k,G}(v)} f(u) \geq 1 \) for each vertex \( v \in V(G) \). A set \( \{f_1, f_2, \ldots, f_d\} \) of signed distance \( k \)-dominating functions on \( G \) with the property that \( \sum_{i=1}^d f_i(v) \leq 1 \) for each \( v \in V(G) \) is called a \({signed\; distance \; k -dominating \;family}\) (of functions) on \( G \). The maximum number of functions in a signed distance \( k \)-dominating family on \( G \) is the \({signed\; distance \; k -domatic\; number}\) of \( G \), denoted by \( d_{k,s}(G) \). Note that \( d_{1,s}(G) \) is the classical signed domatic number \( d_s(D) \). In this paper, we initiate the study of signed distance \( k \)-domatic numbers in graphs and we present some sharp upper bounds for \( d_{k,s}(G) \).

We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacency matrix and automorphism group.

In this paper, we consider a variation of toughness, and prove stronger results for the existence of \([a, b]\)-factors. Furthermore, we show that the results are sharp in some sense.

A decomposition \( \mathcal{D} \) of a graph \( H \) by a graph \( G \) is a partition of the edge set of \( H \) such that the subgraph induced by the edges in each part of the partition is isomorphic to \( G \). The intersection graph \( I(\mathcal{D}) \) of the decomposition \( \mathcal{D} \) has a vertex for each part of the partition and two parts \( A \) and \( B \) are adjacent if and only if they share a common node in \( H \). If \( I(\mathcal{D}) \cong H \), then \( \mathcal{D} \) is an automorphic decomposition of \( H \). If \( n(G) = \chi(H) \) as well, then we say that \( \mathcal{D} \) is a fully automorphic decomposition. In this paper, we examine the question of whether a fully automorphic host will have an even degree of regularity. We also give several examples of fully automorphic decompositions as well as necessary conditions for their existence.

A modular \(k\)-coloring, \(k \geq 2\), of a graph \(G\) without isolated vertices is a coloring of the vertices of \(G\) with the elements in \(\mathbb{Z}_k\) (where adjacent vertices may be colored the same) having the property that for every two adjacent vertices in \(G\) the sums of the colors of their neighbors are different in \(\mathbb{Z}_k\). The minimum \(k\) for which \(G\) has a modular \(k\)-coloring is the modular chromatic number mc\((G)\) of \(G\). It is known that \(2 \leq \text{mc}(T) \leq 3\) for every nontrivial tree \(T\). We present an efficient algorithm that computes the modular chromatic number of a given tree.

In this note, we consider the \(i\)-block intersection graphs (\(i\)-BIG) of a universal friendship \(3\)-hypergraph and show that they are pancyclic for \(i = 1,2\). We also show that the \(1\)-BIG of a universal friendship \(3\)-hypergraph is Hamiltonian-connected.

We study samples \(\Gamma = (\Gamma_1, \ldots, \Gamma_n)\) of length \(n\) where the letters \(\Gamma_i\) are independently generated according to the geometric distribution \(\mathbb{P}(\Gamma_j = i) = pq^{i-1}\), for \(1 \leq j \leq n\), with \(p+q=1\) and \(0<p<1\). An \({up-smooth\; sample}\) \(\Gamma\) is a sample such that \(\Gamma_{i+1}- \Gamma_i \leq 1\). We find generating functions for the probability that a sample of \(n\) geometric variables is up-smooth, with or without a specified first letter. We also extend the up-smooth results to words over an alphabet of \(k\) letters and to compositions of integers. In addition, we study smooth samples \(T\) of geometric random variables, where the condition now is \(|\Gamma_{i+1}- \Gamma_i| \leq 1\).

A Roman dominating function on a graph \(G = (V, E)\) is a function \(f : V \to \{0,1,2\}\) satisfying the condition that every vertex \(u \in V\) for which \(f(u) = 0\) is adjacent to a vertex \(v\) for which \(f(v) = 2\). The weight of a Roman dominating function is the value \(f(V) = \sum_{u \in V} f(u)\). The Roman domination number, \(\gamma_R(G)\), of \(G\) is the minimum weight of a Roman dominating function on \(G\). In this paper, we study those graphs for which the removal of any pair of vertices decreases the Roman domination number. A graph \(G\) is said to be \emph{Roman domination bicritical} or just \(\gamma_R\)-bicritical, if \(\gamma_R(G – \{v,u\}) < \gamma_R(G)\) for any pair of vertices \(v,u \in V\). We study properties of \(\gamma_R\)-bicritical graphs, and we characterize \(\gamma_R\)-bicritical trees and unicyclic graphs.

The van der Waerden number \(W(r, k)\) is the least integer \(N\) such that every \(r\)-coloring of \(\{1, 2, \ldots, N\}\) contains a monochromatic arithmetic progression of length at least \(k\). Rabung gave a method to obtain lower bounds on \(W(2, k)\) based on quadratic residues, and performed computations on all primes no greater than \(20117\). By improving the efficiency of the algorithm of Rabung, we perform the computation for all primes up to \(6 \times 10^7\), and obtain lower bounds on \(W(2, k)\) for \(k\) between \(11\) and \(23\).