Motivated by finding a way to connect the Roman domination number and 2-domination number, which are in general not comparable, we consider a parameter called the Italian domination number (also known as the Roman \((2)\)-domination number). This parameter is bounded above by each of the other two. Bounds on the Italian domination number in terms of the order of the graph are shown. The value of the Italian domination number is studied for several classes of graphs. We also compare the Italian domination number with the 2-domination number.

The 3-sphere regular cellulation conjecture claims that every 2-connected cyclic graph is the 1-dimensional skeleton of a regular cellulation of the 3-dimensional sphere. The conjecture is obviously true for planar graphs. 2-connectivity is a necessary condition for a graph to satisfy such a property. Therefore, the question whether a graph is the 1-dimensional skeleton of a regular cellulation of the 3-dimensional sphere would be equivalent to the 2-connectivity test if the conjecture were proved to be true. On the contrary, it is not even clear whether such a decision problem is computationally tractable.

We introduced a new class of graphs called weakly-split and proved the conjecture for such a class. Hamiltonian, split, complete \( k \)-partite, and matrogenic cyclic graphs are weakly split. In this paper, we introduce another class of graphs for which the conjecture is true. Such a class is a superclass of planar graphs and weakly-split graphs.

The maximum number of internal disjoint paths between any two distinct nodes of faulty enhanced hypercube \( Q_{n,k} (1 \leq k \leq n-1) \) are considered in a more flexible approach. Using the structural properties of \( Q_{n,k} (1 \leq k \leq n-1) \), \( \min(d_{Q_{n,k}-V}(x), d_{Q_{n,k}-V}(y)) \) disjoint paths connecting two distinct vertices \( x \) and \( y \) in an \( n \)-dimensional faulty enhanced hypercube \( Q_{n,k}-V (n \geq 8, k \neq n-2, n-1) \) are conformed when \( |V’| \) is at most \( n-1 \). Meanwhile, it is proved that there exists \( \min(d_{Q_{n,k}-V}(x), d_{Q_{n,k}-V}(y)) \) internal disjoint paths between \( x \) and \( y \) in \( Q_{n,k}-V (n \geq 8, k \neq n-2, n-1) \), under the constraints that (1) The number of faulty vertices is no more than \( 2n-3 \); (2) Every vertex in \( Q_{n,k}-V’ \) is incident to at least two fault-free vertices. This results generalize the results of the faulted hypercube \( FQ_n \), which is a special case of \( Q_{n,k} \), and have improved the theoretical evidence of the fact that \( Q_{n,k} \) has excellent node-fault-tolerance when used as a topology of large-scale computer networks, thus remarkably improving the performance of the interconnect networks.

Given a finite non-empty sequence \( S \) of integers, write it as \( XY^k \), consisting of a prefix \( X \) (which may be empty), followed by \( k \) copies of a non-empty string \( Y \). Then, the greatest such integer \( k \) is called the curling number of \( S \) and is denoted by \( cn(S) \). The notion of curling number of graphs has been introduced in terms of their degree sequences, analogous to the curling number of integer sequences. In this paper, we study the curling number of certain graph classes and graphs associated to given graph classes.

In this paper, we investigate and obtain the properties of higher-order Daehee sequences by using generating functions. In particular, by means of the method of coefficients and generating functions, we establish some identities involving higher-order Daehee numbers, generalized Cauchy numbers, Lah numbers, Stirling numbers of the first kind, unsigned Stirling numbers of the first kind, generalized harmonic polynomials and the numbers \( P(r, n, k) \).

In this paper, we give the sufficient conditions for a graph with large minimum degree to be \( s \)-connected, \( s \)-edge-connected, \( \beta \)-deficient, \( s \)-path-coverable, \( s \)-Hamiltonian and \( s \)-edge-Hamiltonian in terms of spectral radius of its complement.

The maximum number of clues in an \( n \times n \) American-style crossword puzzle grid is explored. Grid constructions provided for all \( n \) are proved to be maximal for all even \( n \). By using mixed integer linear programming, they are verified to be maximal for all odd \( n \leq 49 \). Further, for all \( n \leq 30 \), all maximal grids are provided.

For a graph \( G \), the Merrifield-Simmons index \( i(G) \) is defined as the total number of its independent sets. In this paper, we determine sharp upper and lower bounds on Merrifield-Simmons index of generalized \( \theta \)-graph, which is obtained by subdividing the edges of the multigraph consisting of \( k \) parallel edges, denoted by \( \theta(r_1, r_2, \ldots, r_k) \). The corresponding extremal graphs are also characterized.

For a non-simply connected orthogonal polygon \( T \), assume that \( T = S \setminus (A_1 \cup \ldots \cup A_n) \), where \( S \) is a simply connected orthogonal polygon and where \( A_1, \ldots, A_n \) are pairwise disjoint sets, each the connected interior of an orthogonal polygon, \( A_i \subset S, 1 \leq i \leq n \). If set \( T \) is staircase starshaped, then \( \text{Ker } T = \bigcap \{\text{Ker } (S \setminus A_i) : 1 \leq i \leq n\} \). Moreover, each component of this kernel will be the intersection of the nonempty staircase convex set \( \text{Ker } S \) with a box, providing an easy proof that each of these components is staircase convex. Finally, there exist at most \( (n + 1)^2 \) such components, and the bound \( (n + 1)^2 \) is best possible.

We empirically evaluate the performance of three approximation algorithms for the online bottleneck matching problem. In this matching problem, \( k \) server-vertices lie in a metric space and \( k \) request-vertices that arrive over time each must immediately be permanently assigned to a server-vertex. The goal is to minimize the maximum distance between any request and its assigned server. We consider the naïve \textsc{Greedy} algorithm, as well as \textsc{Permutation} and \textsc{Balance}, each of which were constructed to counter certain challenges in the online problem. We analyze the performance of each algorithm on a variety of data sets, considering each both in the original model, where applicable, and in the resource augmentation setting when an extra server is introduced at each server-vertex. While no algorithm strictly dominates, \textsc{Greedy} frequently performs the best, and thus is recommended due to its simplicity.