For given graphs \( H_1 \) and \( H_2 \), the \({Ramsey\; number}\) \( R(H_1, H_2) \) is the smallest positive integer \( n \) such that if we arbitrarily color the edges of the complete graph \( K_n \) with two colors, 1 (red) and 2 (blue), then there is a monochromatic copy of \( H_1 \) colored with 1 or \( H_2 \) colored with 2.

We show that if \( n \) is even, \( q = \lceil \sqrt{n} \rceil \) is odd, and \( s = n – (q-1)^2 \leq \frac{q}{2} \), then \( R(K_{2,2}, K_{2,n}) \leq n + 2q – 1 \), where \( K_{n,m} \) are complete bipartite graphs. This bound provides the exact value of \( R(K_{2,2}, K_{2,18}) = 27 \). Moreover, we show that \( R(K_{2,2}, K_{2,14}) = 22 \) and \( R(K_{2,2}, K_{2,15}) = 24 \).

A \({red-blue\; coloring}\) of a graph \( G \) is an edge coloring of \( G \) in which every edge is colored red or blue. For a connected graph \( H \) of size at least 2, a \({color \;frame}\) \( F \) of \( H \) is obtained from a red-blue coloring of \( H \) having at least one edge of each color and in which a blue edge is designated as the root edge.

An \( F \)-coloring of a graph \( G \) is a red-blue coloring of \( G \) in which every blue edge of \( G \) is the root edge of a copy of \( F \) in \( G \), and the \( F \)-\({chromatic\; index}\) of \( G \) is the minimum number of red edges in an \( F \)-coloring of \( G \). An \( F \)-coloring of \( G \) is \({minimal}\) if whenever any red edge of \( G \) is changed to blue, then the resulting red-blue coloring of \( G \) is not an \( F \)-coloring of \( G \). The maximum number of red edges in a minimal \( F \)-coloring of \( G \) is the \({upper \; F -chromatic \;index}\) of \( G \).

In this paper, we investigate \( F \)-colorings and \( F \)-chromatic indexes of graphs for all color frames \( F \) of paths of orders 3 and 4.

A set of vertices \( S \) in a graph \( G \) is a dominating set if any vertex of \( G – S \) is adjacent to some vertex in \( S \). The domination number, \( \gamma(G) \), of \( G \) is the minimum cardinality of a dominating set of \( G \).

The subdivision of an edge \( uv \) is the operation of replacing \( uv \) with a path \( uwv \) through a new vertex \( w \). A graph \( G \) is domination critical upon edge subdivision if the domination number increases by subdivision of any edge.

In this paper, we study domination critical graphs upon edge subdivision. We present several properties and bounds for these graphs and then give a constructive characterization of domination critical trees upon edge subdivision.

Let \( G \) be a graph of order \( n \). The \({binding\; number}\) of \( G \) is defined as

\[
\text{bind}(G) := \min \left\{ \frac{|N_G(X)|}{|X|} \mid \emptyset \neq X \subseteq V(G) \text{ and } N_G(X) \neq V(G) \right\}.
\]

A \((g, f)\)-factor is called a connected \((g, f)\)-factor if it is connected. A \((g, f)\)-factor \( F \) is called a Hamilton \((g, f)\)-factor if \( F \) contains a Hamilton cycle. In this paper, several sufficient conditions related to binding number and minimum degree for graphs to have connected \((g, f+1)\)-factors or Hamilton \((g, f)\)-factors are given.

Define an edge \( Q_1Q_2 \) or a triangle \( Q_1Q_2Q_3 \) of a clique graph \( K(G) \) to be weight-\( k \) if \( |Q_1 \cap Q_2| \geq k \) or \( |Q_1 \cap Q_2 \cap Q_3| \geq k \), respectively. A graph \( G \) is shown to be strongly chordal if and only if, for every \( k \geq 1 \), every cycle of weight-\( k \) edges in \( K(G) \) either has a weight-\( k \) chord or is a weight-\( k \) triangle—this mimics the usual definition of chordal graphs. Similarly, trivially perfect graphs have a characterization that mimics a simple characterization of component-complete graphs.

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 \(\lambda > 1\).