The Zarankiewicz number \(z(m, n; s, t)\) is the maximum number of edges in a subgraph of \(K_{m,n}\) that does not contain \(K_{s,t}\) as a subgraph. The \emph{bipartite Ramsey number} \(b(n_1, \ldots, n_k)\) is the least positive integer \(b\) such that any coloring of the edges of \(K_{b,b}\) with \(k\) colors will result in a monochromatic copy of \(K_{n_i,n_i}\) in the \(i\)-th color, for some \(i\), \(1 \leq i \leq k\). If \(n_i = m\) for all \(i\), we denote this number by \(b_k(m)\). In this paper, we obtain the exact values of some Zarankiewicz numbers for quadrilaterals (\(s = t = 2\)), and derive new bounds for diagonal multicolor bipartite Ramsey numbers avoiding quadrilaterals. Specifically, we prove that \(b_4(2) = 19\) and establish new general lower and upper bounds on \(b_k(2)\).

Given non-negative integers \(r\), \(s\), and \(t\), an \({[r, s, t]-coloring}\) of a graph \(G = (V(G), E(G))\) is a function \(c\) from \(V(G) \cup E(G)\) to the color set \(\{0, 1, \ldots, k-1\}\) such that \(|c(v_i) – c(v_j)| \geq r\) for every two adjacent vertices \(v_i\), \(v_j\), \(|c(e_i) – c(e_j)| \geq s\) for every two adjacent edges \(e_i\), \(e_j\), and \(|c(v_i) – c(e_j)| \geq t\) for all pairs of incident vertices \(v_i\) and edges \(e_j\). The [\(r\), \(s\), \(t\)]-chromatic number \(\chi_{r,s,t}(G)\) is the minimum \(k\) such that \(G\) admits an [\(r\), \(s\), \(t\)]-coloring. In this paper, we examine [\(r\), \(s\), \(t\)]-chromatic numbers of fans for every positive integer \(r\), \(s\), and \(t\).

A new hemisystem of the generalized quadrangle \(\mathcal{H}(3, 49)\) admit-
ting the linear group \(PSL_2(7)\) has been found.

A graph is termed Laplacian integral if its Laplacian spectrum comprises integers. Let \(\theta(n_1, n_2, \ldots, n_k)\) be a generalized \(\theta\)-graph (see Figure 1). Denote by \(\mathcal{G}_{k-1}\) the set of \((k-1)\)-cyclic graphs, each containing some generalized \(\theta\)-graph \(\theta(n_1, n_2, \ldots, n_{k})\) as its induced subgraph. In this paper, we establish an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1), from which we identify all Laplacian integral graphs in the class \(\mathcal{G}_{ k-1}\) (Theorem 3.2).

We determine the Ramsey numbers \(R(S_{2,m} K_{2, q})\) for \(m \in \{3, 4, 5\}\) and \(q \geq 2\). Additionally, we obtain \(R(tS_{2, 3}, sK_{2, 2})\) and \(R(S_{2, 3}, sK_{2, 2})\) for \(s \geq 2\) and \(t \geq 1\). Furthermore, we also establish \(R(sK_2, \mathcal{H})\), where \( \mathcal{H}\) is the union of graphs with each component isomorphic to the connected spanning subgraph of \(K_{s} + C_n\), for \(n \geq 3\) and \(s \geq 1\).

For a given set \(M\) of positive integers, a well known problem of Motzkin asks for determining the maximal density \(\mu(M)\) among sets of nonnegative integers in which no two elements differ by an element of \(M\). The problem is completely settled when \(|M| \leq 2\), and some partial results are known for several families of \(M\) for \(|M| \geq 3\),including the case where the elements of \(M\) are in arithmetic progression. We resolve the problem in case of geometric progressions and geometric sequences.

A new Turán-type problem on distances of graphs was introduced by Tyomkyn and Uzzell. In this paper, we focus on the case of distance two. We show that for any positive integer \(n\), a graph \(G\) on \(n\) vertices without three vertices pairwise at distance \(2\) has at most \(\frac{(n-1)^2}{4} + 1\) pairs of vertices at distance \(2\), provided \(G\) has a vertex \(v \in V(G)\) whose neighbors are covered by at most two cliques. This partially answers a conjecture of Tyomkyn and Uzzell [Tyomkyn, M.,Uzzell, A.J.: A new Turdn-type problem on distances of graphs. Graphs Combin. \(29(6), 1927-1942 (2012)\)]..

In the first installment of this series, we proved that for every integer \(a \geq 3\) and every \(m \geq 2a^2 – a + 2\), the \(2\)-color Rado number of \[x_1+x_2+\ldots+x_{m-1}=ax_m\]. is \(\lceil \frac{m-1}{a} \lceil \frac{m-1}{a} \rceil\rceil \). Here, we obtain the best possible improvement of the bound on \(m\). Specifically, we prove that if \(3|a\), then the \(2\)-color Rado number is \(\lceil \frac{m-1}{a} \lceil\frac{m-1}{a} \rceil\rceil \) when \(m \geq 2a + 2\) but not when \(m = 2a+1\), and that if \(3 \nmid\) is composite, then the \(2\)-color Rado number is \(\lceil \frac{m-1}{a}\lceil\frac{m-1}{a}\rceil \rceil \) when \(m \geq 2a + 2\) but not when \(m = 2a + 1\). Additionally, we determine the \(2\)-color Rado number for all \(a \geq 3\) and \(m \geq \frac{a}{3} + 1\).

Let \(G = (V, E)\) be a graph without isolated vertices. A set \(D \subseteq V\) is a paired-dominating set if \(D\) is a dominating set of \(G\) and the induced subgraph \(G[D]\) has a perfect matching. In this paper, we provide a characterization for block graphs with a unique minimum paired-dominating set. Furthermore, we also establish a constructive characterization for trees with a unique minimum paired-dominating set.

Estimates of the choice numbers and the Ohba numbers of the complete multipartite graphs \(K(m, n, 1, \ldots, 1)\) and \(K(m, n, 2, \ldots, 2)\) are provided for various values of \(m \geq n \geq 1\). The Ohba number of a graph \(G\) is the smallest integer \(n\) such that \(\operatorname{ch}(G \vee K_n) = \chi(G \vee K_n)\).