In this paper, we study the distribution on \([k]^n\) for the parameter recording the number of indices \(i \in [n-1]\) within a word \(w=w_1\cdots w_n\) such that \(|w_{i+1}-w_i|\ \leq 1\) and compute the corresponding (bivariate) generating function. A circular version of the problem wherein one considers whether or not \(|w_n-w_1|\ \leq 1\) as well is also treated. As special cases of our results, one obtains formulas involving staircase and Hertzsprung words in both the linear and circular cases. We make use of properties of special matrices in deriving our results, which may be expressed in terms of Chebyshev polynomials. A generating function formula is also found for the comparable statistic on finite set partitions with a fixed number of blocks represented sequentially.

Let \(G=(V,E)\) be a graph of order \(n\) without isolated vertices. A bijection \(f\colon V\rightarrow \{1,2,\dots,n\}\) is called a local distance antimagic labeling, if \(w(u)\not=w(v)\) for every edge \(uv\) of \(G\), where \(w(u)=\sum_{x\in N(u)}f(x)\). The local distance antimagic chromatic number \(\chi_{ld}(G)\) is defined to be the minimum number of colors taken over all colorings of \(G\) induced by local distance antimagic labelings of \(G\). The concept of Generalized Mycielskian graphs was introduced by Stiebitz [20]. In this paper, we study the local distance antimagic labeling of the Generalized Mycielskian graphs.

A graph is called \(t\)-existentially closed (\(t\)-e.c.) if it satisfies the following adjacency property: for every \(t\)-element subset \(A\) of the vertices, and for every subset \(B \subseteq A\), there exists a vertex \(x \in A\) that is adjacent to all vertices in \(B\) and to none of the vertices in \(A \setminus B\). A \(t\)-e.c. graph is critical if removing any single vertex results in a graph that is no longer \(t\)-e.c. This paper investigates \(2\)-e.c. critical Cayley graphs and vertex-transitive graphs, providing explicit constructions of \(2\)-e.c. critical Cayley graphs on cyclic groups. It is shown that a \(2\)-e.c. critical Cayley graph (as well as vertex-transitive graphs) of order \(n\) exists if and only if \(n \geq 9\) and \(n \notin \{10, 11, 14\}\). Additionally, this paper examines the numbers of \(2\)-e.c. (critical) vertex-transitive graphs among all vertex-transitive graphs for small orders, and presents detailed observations on some \(2\)-e.c. and \(3\)-e.c. vertex-transitive graphs.

For a family \(\mathcal F\) of graphs, a graph \(G\) is said to be \(\mathcal F\)-free if \(G\) contains no member of \(\mathcal F\) as an induced subgraph. We let \(\mathcal G_{3}(\mathcal F)\) be the family of \(3\)-connected \(\mathcal F\) -free graphs. Let \(P_{n}\) and \(C_{n}\) denote the path and the cycle of order \(n\), respectively. Let \(T_{0}\) be the tree of order nine obtained by joining a pendant edge to the central vertex of \(P_{7}\). Let \(T_{1}\) and \(T_{2}\) be the trees of order ten obtained from \(T_{0}\) by joining a new vertex to a vertex of \(P_{7}\) adjacent to an endvertex, and to a vertex of \(P_{7}\) adjacent to the central vertex, respectively. We show that \(\mathcal G_{3}(\{C_{3}, C_{4}, T_{1}\})\) and \(\mathcal G_{3}(\{C_{3}, C_{4}, T_{2}\})\) are finite families.

We model each 4\(\mathrm{\times}\)4 magic square by encoding its 16 integers as magnitudes of repelling positive point charges on a fixed 2D lattice, evolved under Coulomb forces with linear damping and a harmonic pinning to anchor sites. We simulate all 880 magic squares and compare them with equally sized ensembles of random permutations of \(\{1,{\dots},16\}\). Three readouts differentiate the ensembles. (i) Final positions: magic cases form sixteen tight, index-specific clusters on an annulus, whereas random cases show broader arcs and central accumulation. (ii) Displacement–correlation structure: across cases, many pair-of-pairs of inter-index displacements in magic squares are near-linearly dependent; the random ensemble exhibits only moderate relationships, with |r| and R\(^2\) distributions shifted to weaker correlation. (iii) Center potential: the Coulomb-type potential at the geometric center collapses to a single value at the anchors for all magic squares and remains narrowly distributed after dynamics, while random squares remain broad. Sensitivity analysis reveals a broad damping–stiffness region with high convergence, indicating that the results are robust to parameter choice.

Tolerance graphs, introduced by M. C. Golumbic and C. L. Monma in 1982, are a generalization of interval graphs. In this paper, we propose an application of coloring of tolerance graphs together with machine learning, in particular supervised learning, in solving problems of airport gate assignment during a pandemic of an airborne disease. The idea is to minimize a contact between passengers at the gates, in order to slow down a spread of the disease. This application includes calculating the chromatic number of a graph and finding a clique of a given size in the graph. As a result, we obtain the minimum number of gates needed under the given assumptions and the corresponding gate assignment. Further, we propose an application of list coloring, a generalization of a coloring, of tolerance graphs in solving these problems. Besides the theoretical approach, the corresponding algorithms are given. The algorithms developed may take into account several parameters, such as the number of passengers on a flight, the number of newly infected people per 1000 inhabitants. A similar approach can be taken for classroom assignment during a pandemic, scheduling meetings, etc.

This paper addresses the enumeration of two types of zero-sum sequences. The first type is defined by constraints on both the number of terms and the values they may assume; the second type is constrained by the number of terms and the total absolute value of the sequence. For both cases, exact enumeration formulas are derived in terms of restricted partition functions, and asymptotic estimates are obtained. In the final section, an algebraic analogue of restricted compositions is introduced, and its enumerative properties are analyzed.

A Fibonacci cordial labeling of a graph \(G\) is an injective function \(f: V(G) \rightarrow \{F_0, F_1, \dots,\\ F_n\}\), where \(F_i\) denotes the \(i^{\text{th}}\) Fibonacci number, such that the induced edge labeling \(f^*: E(G) \rightarrow \{0,1\}\), given by \(f^*(uv) = (f(u) + f(v))\) \((\bmod\ 2)\), satisfies the balance condition \(|e_f(0) – e_f(1)| \le 1\). Here, \(e_f(0)\) and \(e_f(1)\) represent the number of edges labeled 0 and 1, respectively. A graph that admits such a labeling is termed a Fibonacci cordial graph. In this paper, we investigate the existence and construction of Fibonacci cordial labelings for several families of graphs, including Generalized Petersen graphs, open and closed helm graphs, joint sum graphs, and circulant graphs of small order. New results and examples are presented, contributing to the growing body of knowledge on graph labelings inspired by numerical sequences.

An S-packing k-coloring of a graph \(G\) (with \(S=(s_1,s_2,\dots)\) is a non-decreasing sequence of positive integers) is a mapping \(f\) from \(V(G)\) to \(\lbrace 1,\dots , k \rbrace\) (the set of colors) such that for every two distincts vertices \(x\) and \(y\) in \(V(G)\) with \(f(x)=f(y)=i\) the distance between \(x\) and \(y\) in \(G\) is bigger than \(s_i\). The S-packing chromatic number \(\chi_S (G)\) of \(G\) is the smallest integer \(k\) such that \(G\) has an S-packing k-coloring. Given a set \(D\subset \mathbb{N}^*\), a distance graph \(G(\mathbb{Z}, D)\) with distance set \(D\) is a graph with vertex set \(\mathbb{Z}\) and two distincts vertices \(u\) and \(v\) are adjacents if \(| u-v | \in D\). In this paper, for \(S=(s,s+1,s+1,\dots)\) with \(s \geq \left\lceil \frac{t}{2} \right\rceil\) we give a lower bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\), and a lower bound of \(\chi_d (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) with \(d \geq \left\lceil \frac{t}{2} \right\rceil\), for \(S=(s_1,s_2,\dots, s_i,a,a,\dots)\) with \(a \geq \max( 1 , t-2 )\) we give an upper bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\), and we determine the exact values of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) and also of \(\chi_d (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) for \(s\geq \max (\left\lceil \frac{t}{2} \right\rceil, t-3)\) and \(d \geq \max (\left\lceil \frac{t}{2} \right\rceil , t-2)\). And we give a lower and an upper bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) for \(S=(1,s,s,\dots)\) with conditions on \(s\) and \(t\), which in the cases \(s\geq \max (t-2,\left\lceil \frac{t}{2}\right\rceil)\) we determine the exact values of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\).

For a graph \(G=(V,E)\), a pair of vertex disjoint sets \(A_{1}\) and \(A_{2}\) form a connected coalition of \(G\), if \(A_{1}\cup A_{2}\) is a connected dominating set, but neither \(A_{1}\) nor \(A_{2}\) is a connected dominating set. A connected coalition partition of \(G\) is a partition \(\Phi\) of \(V(G)\) such that each set in \(\Phi\) either consists of only a singe vertex with the degree \(\mid V(G)\mid-1\), or forms a connected coalition of \(G\) with another set in \(\Phi\). The connected coalition number of \(G\), denoted by \(CC(G)\), is the largest possible size of a connected coalition partition of \(G\). In this paper, we characterize graphs that satisfy \(CC(G)=2\). Moreover, we obtain the connected coalition number for unicycle graphs and for the corona product and join of two graphs. Finally, we give a lower bound on the connected coalition number of the Cartesian product and the lexicographic product of two graphs.