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.

In this paper, we study the total domatic partition problem for bipartite graphs, split graphs, and graphs with balanced adjacency matrices. We show that the total domatic partition problem is NP-complete for bipartite graphs and split graphs, and show that the total domatic partition problem is polynomial-time solvable for graphs with balanced adjacency matrices. Furthermore, we show that the total domatic partition problem is linear-time solvable for any chordal bipartite graph \( G \) if a \( \Gamma \)-free form of the adjacency matrix of \( G \) is given.

A graphic sequence \( \pi = (d_1, d_2, \ldots, d_n) \) is said to be potentially \( K_{1^3,4} \)-graphic if there is a realization of \( \pi \) containing \( K_{1^3,4} \) as a subgraph, where \( K_{1^3,4} \) is the \( 1 \times 1 \times 1 \times 4 \) complete 4-partite graph. In this paper, we characterize the graphic sequences potentially \( K_{1^3,4} \)-graphic and the result is simple. In addition, we apply this characterization to compute the values of \( \sigma( K_{1^3,4}, n) \).

We show, using a hybrid analysis/linear algebra argument, that the diagonal vector of an infinite symmetric matrix over \(\mathbb{Z}_{2}\) is contained in the range of the matrix. We apply this result to an extension, to the countably infinite case, of the Lights Out problem.