For a nonempty graph \(G = (V(G), E(G))\), a signed cycle dominating function on \(G\) is introduced by Xu in 2009 as a function \(f : E(G) \to \{1, -1\}\) such that \(\sum_{e \in E(C)} f(e) \geq 1\) for any induced cycle \(C\) of \(G\). A set \(\{f_1, f_2, \dots, f_d\}\) of distinct signed cycle dominating functions on \(G\) with the property that \(\sum_{i=1}^{d} f_i(e) \leq 1\) for each \(e \in E(G)\), is called a signed cycle dominating family (of functions) on \(G\). The maximum number of functions in a signed cycle dominating family on \(G\) is the signed cycle domatic number of \(G\), denoted by \(d’_{sc}(G)\). In this paper, we study the signed cycle domatic numbers in graphs and present sharp bounds for \(d’_{sc}(G)\). In addition, we determine the signed cycle domatic number of some special graphs.

Using partition theoretic methods we combinatorially interpret the four Ae Rogers—Ramanujan identities of Andrews, Schilling and Wamaar.

Let \(p > 165\) be a prime and let \(G\) be a cyclic group of order \(p\). Let \(S\) be a minimal zero-sum sequence with elements over \(G\), i.e., the sum of elements in \(S\) is zero, but no proper nontrivial subsequence of \(S\) has sum zero. We call \(S\) unsplittable, if there do not exist \(g \in S\) and \(x, y \in G\) such that \(g = x + y\) and \(Sg^{-1}x y\) is also a minimal zero-sum sequence. In this paper, we determine the structure of \(S\) which is an unsplittable minimal zero-sum sequence of length \(\frac{p-1}{2}\) or \(\frac{p-3}{2}\). Furthermore, if \(S\) is a minimal zero-sum sequence with \(|S| \geq \frac{p-3}{2}\), then \(ind(S) \leq 2\).

For two given graphs \(G_1\) and \(G_2\), the Ramsey number \(R(G_1, G_2)\) is the smallest integer \(x\) such that for any graph \(G\) of order \(n\), either \(G\) contains \(G_1\) or the complement of \(G\) contains \(G_2\). In this paper, we study a large class of trees \(T\) as studied by Cockayne in [3], including paths and trees which have a vertex of degree one adjacent to a vertex of degree two, as special cases. We evaluate some \(R(T’_m, B_m)\), where \(T’_n \in \mathbb{T}\) and \(B_m\) is a book of order \(m+2\). Besides, some bounds for \(R(T’_n, B_n)\) are obtained.

Graceful labeling of graphs is used in radar codes. In this work, we introduce a new version of gracefulness, which we call edge-even graceful labeling of graphs. We establish a necessary and sufficient condition for edge-even graceful labeling of path graphs \(P_n\), cycle graphs \(C_n\), and star graphs \(K_{1,n}\). We also prove some necessary and sufficient conditions for some path and cycle-related graphs, namely, Friendship, Wheel, Double wheel, and Fan graphs.

The Hamiltonian problem is a classical problem in graph theory. Most of the research on the Hamiltonian problem is looking for sufficient conditions for a graph to be Hamiltonian. For a vertex \(v\) of a graph \(G\), Zhu, Li, and Deng introduced the concept of implicit degree \(id(v)\), according to the degrees of its neighbors and the vertices at distance \(2\) with \(v\) in \(G\). In this paper, we will prove that: Let \(G\) be a \(2\)-connected graph on \(n \geq 3\) vertices. If the maximum value of the implicit degree sums of \(2\) vertices in \(S\) is more than or equal to \(n\) for each independent set \(S\) with \(\kappa(G) + 1\) vertices, then \(G\) is Hamiltonian.

Let \((d_1, d_2, \dots, d_n)\) be a sequence of positive integers with \(n-1 \geq d_1 \geq d_2 \geq \dots \geq d_n\). We give a characterization of \((d_1, d_2, \dots, d_n)\) that is the degree sequence of a graph with cyclomatic number \(k\). This simplifies the characterization of Erdős-Gallai.

We explore new combinatorial properties of overpartitions, which are natural generalizations of integer partitions. Building on recent work, we state general combinatorial identities between standard partition, overpartition, and regular partition functions. We provide both generating function and bijective proofs. We also prove congruences for certain overpartition functions combinatorially.

Let \(G\) be a simple graph on \(n\) vertices. The Laplacian Estrada index of \(G\) is defined as \(LEE(G) = \sum_{i=1}^{n} e^{\mu_i}\), where \(\mu_1, \mu_2, \dots, \mu_n\) are the Laplacian eigenvalues of \(G\). In this paper, threshold graphs on \(n\) vertices and \(m\) edges having maximal and minimal Laplacian Estrada index are determined, respectively.

In this paper, formulas of the resistance distance for the arbitrary two-vertex resistance of \(G\), \(H = G_1 \boxdot G_2\) and \(G_1 \boxminus G_2\) in the electrical networks are obtained in a much simpler way. Furthermore, \(K_f(G_1 \boxdot G_2)\) and \(K_f(G_1 \boxminus G_2)\) can be expressed as a combination of \(K_f(G_1)\) and \(K_f(G_2)\).