By the partial fraction decomposition method, we establish a \(q\)-harmonic sum identity with multi-binomial coefficient, from which we can derive a fair number of harmonic number identities.

A fall \(k\)-coloring of a graph \(G\) is a proper \(k\)-coloring of \(G\) such that each vertex of \(G\) sees all \(k\) colors on its closed neighborhood. We denote \(\text{Fall}(G)\) the set of all positive integers \(k\) for which \(G\) has a fall \(k\)-coloring. In this paper, we study fall colorings of the lexicographic product of graphs and the categorical product of graphs. Additionally, we show that for each graph \(G\), \(\text{Fall}(M(G)) = \emptyset\), where \(M(G)\) is the Mycielskian of the graph \(G\). Finally, we prove that for each bipartite graph \(G\), \(\text{Fall}(G^c) \subseteq \{\chi(G^c)\}\) and it is polynomial time to decide whether \(\text{Fall}(G^c) = \{\chi(G^c)\}\) or not.

In this paper, we first provide two necessary conditions for a graph \(G \) to be \(E_k\)-cordial, then we prove that every \(P_n(n \geq 3)\) is \(E_p\)-cordial if \(p\) is odd. In the end, we discuss the \(E_2\)-cordiality of a graph \)G\) under the condition that some subgraph of \(G\) has a \(1\)-factor.

In this paper, we consider the problem of determining the structure of a minimal critical set in a latin square \(L\) representing the elementary abelian \(2\)-group of order \(8\).

In this paper, the first two (resp. four) largest signless Laplacian spectral radii together with the corresponding graphs in the class of bicyclic (resp. tricyclic) graphs of order n are determined, and the first two (resp. four) largest signless Laplacian spreads together with the corresponding graphs in the class of bicyclic (resp. tricyclic) graphs of order \(n\) are identified.

An edge-magic total labeling of a graph \(G\) is a one-to-one map \(\lambda\) from \(V(G) \cup E(G)\) onto the integers \(\{1, 2, \ldots, |V(G) \cup E(G)|\}\) with the property that there exists an integer constant \(c\) such that \(\lambda(x) + \lambda(x,y) + \lambda(y) = c\) for any \((x, y) \in E(G)\). If \(\lambda(V(G)) = \{1, 2, \ldots, |V(G)|\}\), then the edge-magic total labeling is called super edge-magic total labeling. In this paper, we formulate super edge-magic total labeling on subdivisions of stars \(K_{1,p}\), for \(p \geq 5\).

In this paper, we briefly survey Euler’s works on identities connected with his famous Pentagonal Number Theorem. We state a partial generalization of his theorem for partitions with no part exceeding an identified value \(k\), along with some identities linking total partitions to partitions with distinct parts under the above constraint. We derive both recurrence formulas and explicit forms for \(\Delta_n(m)\), where \(\Delta_n(m)\) denotes the number of partitions of \(m\) into an even number of distinct parts not exceeding \(n\), minus the number of partitions of \(m\) into an odd number of distinct parts not exceeding \(n\). In fact, Euler’s Pentagonal Number Theorem asserts that for \(m \leq n\), \(\Delta_n(m) = \pm 1\) if \(m\) is a Pentagonal Number and \(0\) otherwise. Finally, we establish two identities concerning the sum of bounded partitions and their connection to prime factors of the bound integer.

Consider the following one-person game: let \(S = {F_1, F_2,\ldots, F_r}\) be a family of forbidden graphs. The edges of a complete graph are randomly shown to the Painter one by one, and he must color each edge with one of \(r\) colors when it is presented, without creating some fixed monochromatic forbidden graph \(F\); in the \(i\)-th color. The case of all graphs \(F\); being cycles is studied in this paper. We give a lower bound on the threshold function for online \(S\)-avoidance game,which generalizes the results of Marciniszyn, Spdhel and Steger for the symmetric case. [Combinatorics, Probability and Computing, Vol. \(18, 2009: 271-300.\)]

Given positive integers \(n\), \(k\), and \(m\), the \((n,k)\)-th \(m\)-restrained Stirling number of the first kind is the number of permutations of an \(n\)-set with \(k\) disjoint cycles of length \(\leq m\). By inverting the matrix consisting of the \((n,k)\)-th \(m\)-restrained Stirling number of the first kind as the \((n+1,k+1)\)-th entry, the \((n,k)\)-th \(m\)-restrained Stirling number of the second kind is defined. In this paper, we study the multi-restrained Stirling numbers of the first and second kinds to derive their explicit formulae, recurrence relations, and generating functions. Additionally, we introduce a unique expansion of multi-restrained Stirling numbers for all integers \(n\) and \(k\), and a new generating function for the Stirling numbers of the first kind.

Employing \(q\)-commutive structures, we develop binomial analysis and combinatorial applications induced by an important operator in
analogue Fourier analysis associated with well-known \(q\)-series of L.J. Rogers.