Let \( G = (V_1, V_2; E) \) be a bipartite graph with \( |V_1| = |V_2| = 2k \), where \( k \) is a positive integer. It is proved that if \( d(x) + d(y) \geq 3k \) for every pair of nonadjacent vertices \( x \in V_1 \), \( y \in V_2 \), then \( G \) contains \( k \) independent quadrilaterals.

A set \( S \) of vertices of a graph \( G \) is geodetic if every vertex in \( V(G) \setminus S \) is contained in a shortest path between two vertices of \( S \). The geodetic number \( g(G) \) is the minimum cardinality of a geodetic set of \( G \). The geodomatic number \( d_g(G) \) of a graph \( G \) is the maximum number of elements in a partition of \( V(G) \) into geodetic sets.

In this paper, we determine \( d_g(G) \) for some family of graphs, and we present different bounds on \( d_g(G) \). In particular, we prove the following Nordhaus-Gaddum inequality, where \( \overline{G} \) is the complement of the graph \( G \). If \( G \) is a graph of order \( n \geq 2 \), then
\[
d_g(G) + d_g(\overline{G}) \leq n
\]
with equality if and only if \( n = 2 \).

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 \( \overline{\zeta_n^1} \).