We define the \( (i, j) \)-liars’ domination number of \( G \), denoted by \( LR(i, j)(G) \), to be the minimum cardinality of a set \( L \subseteq V(G) \) such that detection devices placed at the vertices in \( L \) can precisely determine the set of intruder locations when there are between 1 and \( i \) intruders and at most \( j \) detection devices that might “lie”.

We also define the \( X(c_1, c_2, \ldots, c_t, \ldots) \)-domination number, denoted by \( \gamma _{X(c_1, c_2, \ldots, c_t, \ldots)}(G) \), to be the minimum cardinality of a set \( D \subseteq V(G) \) such that, if \( S \subseteq V(G) \) with \( |S| = k \), then \( |(\bigcup_{v \in S} N[v]) \cap D| \geq c_k \). Thus, \( D \) dominates each set of \( k \) vertices at least \( c_k \) times making \( \gamma_{X(c_1, c_2, \ldots, c_t, \ldots)}(G) \) a set-sized dominating parameter. We consider the relations between these set-sized dominating parameters and the liars’ dominating parameters.

For Cauchy numbers of the first kind \( \{a_n\}{n \geq 0} \) and Cauchy numbers of the second kind \( \{b_n\}{n \geq 0} \), this paper focuses on the log-convexity of some sequences related to \( \{a_n\}{n \geq 0} \) and \( \{b_n\}{n \geq 0} \). For example, we discuss log-convexity of \( \{n|a_n| – |a_{n+1}|\}{n \geq 1} \), \( \{b{n+1} – nb_n\}{n \geq 1} \), \( \{n|a_n|\}{n \geq 1} \), and \( \{(n + 1)b_n\}_{n \geq 0} \). In addition, we investigate log-balancedness of some sequences involving \( a_n \) (or \( b_n \)).

Let \( G \) be a graph. We define the distance \( d \) pebbling number of \( G \) to be the smallest number \( s \) such that if \( s \) pebbles are placed on the vertices of \( G \), then there must exist a sequence of pebbling moves which takes a pebble to a vertex which is at a distance of at least \( d \) from its starting point. In this article, we evaluate the distance \( d \) pebbling numbers for a directed cycle graph with \( n \) vertices.

Let \( G \) be a \( k \)-connected (\( k \geq 2 \)) graph of order \( n \). If \( \gamma(G^c) \geq n – k \), then \( G \) is Hamiltonian or \( K_k \vee K_{k+1}^c \), where \( \gamma(G^c) \) is the domination number of the complement of the graph \( G \).

An \emph{Italian dominating function} on a digraph \( D \) with vertex set \( V(D) \) is defined as a function \( f : V(D) \to \{0, 1, 2\} \) such that every vertex \( v \in V(D) \) with \( f(v) = 0 \) has at least two in-neighbors assigned 1 under \( f \) or one in-neighbor \( w \) with \( f(w) = 2 \). The weight of an Italian dominating function is the sum \( \sum_{v \in V(D)} f(v) \), and the minimum weight of an Italian dominating function \( f \) is the \emph{Italian domination number}, denoted by \( \gamma_I(D) \). We initiate the study of the Italian domination number for digraphs, and we present different sharp bounds on \( \gamma_I(D) \). In addition, we determine the Italian domination number of some classes of digraphs. As applications of the bounds and properties on the Italian domination number in digraphs, we give some new and some known results of the Italian domination number in graphs.

A hypergraph \( H \) with vertex set \( V \) and edge set \( E \) is called bipartite if \( V \) can be partitioned into two subsets \( V_1 \) and \( V_2 \) such that \( e \cap V_1 \neq \phi \) and \( e \cap V_2 \neq \phi \) for any \( e \in E \). A bipartite self-complementary 3-uniform hypergraph \( H \) with partition \( (V_1, V_2) \) of a vertex set \( V \) such that \( |V_1| = m \) and \( |V_2| = n \) exists if and only if either (i) \( m = n \) or (ii) \( m \neq n \) and either \( m \) or \( n \) is congruent to 0 modulo 4 or (iii) \( m \neq n \) and both \( m \) and \( n \) are congruent to 1 or 2 modulo 4.

In this paper we prove that, there exists a regular bipartite self-complementary 3-uniform hypergraph \( H(V_1, V_2) \) with \( |V_1| = m, |V_2| = n, m + n > 3 \) if and only if \( m = n \) and \( n \) is congruent to 0 or 1 modulo 4. Further we prove that, there exists a quasi-regular bipartite self-complementary 3-uniform hypergraph \( H(V_1, V_2) \) with \( |V_1| = m, |V_2| = n, m + n > 3 \) if and only if either \( m = 3, n = 4 \) or \( m = n \) and \( n \) is congruent to 2 or 3 modulo 4.

Neighborhood-prime labeling is a variation of prime labeling. A labeling \( f : V(G) \to [|V(G)|] \) is a neighborhood-prime labeling if for each vertex \( v \in V(G) \) with degree greater than 1, the greatest common divisor of the set of labels in the neighborhood of \( v \) is 1. In this paper, we introduce techniques for finding neighborhood-prime labelings based on the Hamiltonicity of the graph, by using conditions on possible degrees of vertices, and by examining a neighborhood graph. In particular, classes of graphs shown to be neighborhood-prime include all generalized Petersen graphs, grid graphs of any size, and lobsters given restrictions on the degree of the vertices. In addition, we show that almost all graphs and almost all regular graphs have neighborhood-prime, and we find all graphs of this type.

Let \( A(n, d, w) \) denote the maximum size of a binary code with length \( n \), minimum distance \( d \), and constant weight \( w \). The following lower bounds are here obtained in computer searches for codes with prescribed automorphisms: \( A(16, 4, 6) \geq 624 \), \( A(19, 4, 8) \geq 4698 \), \( A(20, 4, 8) \geq 7830 \), \( A(21, 4, 6) \geq 2880 \), \( A(22, 6, 6) \geq 343 \), \( A(24, 4, 5) \geq 1920 \), \( A(24, 6, 9) \geq 3080 \), \( A(24, 6, 11) \geq 5376 \), \( A(24, 6, 12) \geq 5558 \), \( A(25, 4, 5) \geq 2380 \), \( A(25, 6, 10) \geq 6600 \), \( A(26, 4, 5) \geq 2816 \), and \( A(27, 4, 5) \geq 3456 \).

For a finite simple graph \( G \), say \( G \) is of dimension \( n \), and write \( \text{dim}(G) = n \), if \( n \) is the smallest integer such that \( G \) can be represented as a unit-distance graph in \( \mathbb{R}^n \). Define \( G \) to be \emph{dimension-critical} if every proper subgraph of \( G \) has dimension less than \( G \). In this article, we determine exactly which complete multipartite graphs are dimension-critical. It is then shown that for each \( n \geq 2 \), there is an arbitrarily large dimension-critical graph \( G \) with \( \text{dim}(G) = n \). We close with a few observations and questions that may aid in future work.

Let \( G \) be a simple and finite graph. A graph is said to be decomposed into subgraphs \( H_1 \) and \( H_2 \) which is denoted by \( G = H_1 \oplus H_2 \), if \( G \) is the edge disjoint union of \( H_1 \) and \( H_2 \). If \( G = H_1 \oplus H_2 \oplus \cdots \oplus H_k \), where \( H_1, H_2, \ldots, H_k \) are all isomorphic to \( H \), then \( G \) is said to be \( H \)-decomposable. Furthermore, if \( H \) is a cycle of length \( m \), then we say that \( G \) is \( C_m \)-decomposable and this can be written as \( C_m \mid G \). Where \( G \times H \) denotes the tensor product of graphs \( G \) and \( H \), in this paper, we prove that the necessary conditions for the existence of \( C_6 \)-decomposition of \( K_m \times K_n \) are sufficient. Using these conditions it can be shown that every even regular complete multipartite graph \( G \) is \( C_6 \)-decomposable if the number of edges of \( G \) is divisible by 6.