For given finite simple graphs \( F \) and \( G \), the Ramsey number \( R(F, G) \) is the minimum positive integer \( n \) such that for every graph \( H \) of order \( n \), either \( H \) contains \( F \) or the complement of \( H \) contains \( G \). In this note, with the help of computer, we get that \(R(C_5, W_6) = 13, \quad R(C_5, W_7) = 15, \quad R(C_5, W_8) = 17\),\(R(C_6, W_6) = 11, \quad R(C_6, W_7) = 16, \quad R(C_6, W_8) = 13\),\(R(C_7, W_6) = 13 \quad \text{and} \quad R(C_7, W_8) = 17\).

A \((p,q)\)-graph is said to be a permutation graph if there exists a bijection function \( f: V(G) \to \{1, 2, \ldots, p\} \) such that the induced edge function \( h_f: E(G) \to \mathbb{N} \) is defined as follows:
\[
h_f(x_i, x_j) =
\begin{cases}
{}^{f(x_i)}P_{f(x_j)}, & \text{if } f(x_j) < f(x_i); \\ {}^{f(x_j)}P_{f(x_i)}, & \text{if } f(x_i) < f(x_j). \end{cases} \] In this paper, we investigate the permutation labelings of wheel-related graphs.

Determining whether or not a graph has an efficient dominating set (equivalently, a perfect code) is an NP-complete problem. Here we present a polynomial time algorithm to decide if a given simplicial graph has an efficient dominating set. However, the efficient domination number decision problem is NP-complete for simplicial graphs.

The purpose of this note is to give two binomial sums with generalized Fibonacci sequences. These results generalize two binomial sums by Kilic and Ionascu in The Fibonacci Quarterly, 48.2(2010), 161-167.

Let \( G \) be a connected graph of size at least 2 and \( c: E(G) \to \{0, 1, \ldots, k-1\} \) an edge coloring (or labeling) of \( G \) using \( k \) colors (where adjacent edges may be assigned the same color). For each vertex \( v \) of \( G \), the color code of \( v \) with respect to \( c \) is the \( k \)-tuple \( \text{code}(v) = (a_0, a_1, \ldots, a_{k-1}) \), where \( a_i \) is the number of edges incident with \( v \) that are labeled \( i \) (for \( 0 \leq i \leq k-1 \)). The labeling \( c \) is called a detectable labeling if distinct vertices in \( G \) have distinct color codes. The value \( \text{val}(c) \) of a detectable labeling \( c \) of a graph \( G \) is the sum of the colors assigned to the edges in \( G \). The total detection number \( \text{td}(G) \) of \( G \) is defined by \( \text{td}(G) = \min\{\text{val}(c)\} \), where the minimum is taken over all detectable labelings \( c \) of \( G \). Thus, if \( G \) is a connected graph of size \( m \geq 2 \), then \( 1 \leq \text{td}(G) \leq \binom{m}{2} \). We present characterizations of all connected graphs \( G \) of size \( m \geq 2 \) for which \( \text{td}(G) \in \{1, \binom{m}{2}\} \). The total detection numbers of complete graphs and cycles are also investigated.

In this paper we prove that every planar graph without \(5\)- and \(8\)-cycles and without adjacent triangles is \(3\)-colorable.

A new construction of authentication codes with arbitration using singular pseudo-symplectic geometry on finite fields is given. Some parameters and the probabilities of success for different types of deceptions are computed.

Two graphs are defined to be adjointly equivalent if their complements are chromatically equivalent. By \( h(G,x) \) and \( P(G,\lambda) \) we denote the adjoint polynomial and the chromatic polynomial of graph \( G \), respectively. A new invariant of graph \( G \), which is the fifth character \( R_5(G) \), is given in this paper. Using this invariant and the properties of the adjoint polynomials, we firstly and completely determine the adjoint equivalence class of the graph \( \zeta_n^1 \). According to the relations between \( h(G,x) \) and \( P(G,\lambda) \), we also simultaneously determine the chromatic equivalence class of \( \overline{\zeta_n^1} \).

In this paper, we discuss the properties of a class of generalized harmonic numbers \( H_{n,r} \). Using Riordan arrays and generating functions, we establish some identities involving \( H_{n,r} \). Furthermore, we investigate certain sums related to harmonic polynomials \( H_n(z) \). In particular, using the Riordan array method, we explore interesting relationships between these polynomials, the generating Stirling polynomials, the Bernoulli polynomials, and the Cauchy polynomials. Finally, we obtain the asymptotic expansion of certain sums involving \( H_{n,r} \).

We prove that \( F_v(3,5;6) = 16 \), which solves the smallest open case of vertex Folkman numbers of the form \( F_v(3, k; k+1) \). The proof uses computer algorithms.