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.

In [H. Ngo, D. Du, New constructions of non-adaptive and error-tolerance
pooling designs, Discrete Math. \(243 (2002) 167-170\)], by using subspaces
in a vector space Ngo and Du constructed a family of well-known pooling
designs. In this paper, we construct a family of pooling designs by using
bilinear forms on subspaces in a vector space, and show that our design and
Ngo-Du’s design have the same error-tolerance capability but our design is
more economical than Ngo-Du’s design under some conditions.

A transverse Steiner quadruple system \((TSQS)\) is a triple \((X, \mathcal{H}, \mathcal{B})\) where \(X\) is a \(v\)-element set of points, \(\mathcal{H} = \{H_1, H_2, \ldots, H_r\}\) is a partition of \(X\) into holes, and \(\mathcal{B}\) is a collection of transverse \(4\)-element subsets with respect to \(\mathcal{H}\), called blocks, such that every transverse \(3\)-element subset is in exactly one block. In this article, we study transverse Steiner quadruple systems with \(r\) holes of size \(g\) and \(1\) hole of size \(u\). Constructions based on the use of \(s\)-fans are given, including a construction for quadrupling the number of holes of size \(g\). New results on systems with \(6\) and \(11\) holes are obtained, and constructions for \(\text{TSQS}(x^n(2n)^1)\) and \(\text{TSQS}(4^n2^1)\) are provided.

Let \(G\) be a subgraph of the complete graph \(K_{r+1}\) on \(r+1\) vertices, and let \(K_{r+1} – E(G)\) be the graph obtained from \(K_{r+1}\) by deleting all edges of \(G\). A non-increasing sequence \(\pi = (d_1, d_2, \ldots, d_n)\) of nonnegative integers is said to be potentially \(K_{r+1} – E(G)\)-graphic if it is realizable by a graph on \(n\) vertices containing \(K_{r+1} – E(G)\) as a subgraph. In this paper, we give characterizations for \(\pi = (d_1, d_2, \ldots, d_n)\) to be potentially \(K_{r+1} – E(G)\)-graphic for \(G = 3K_2, K_3, P_3, K_{1,3}\), and \(K_2 \cup P_2\), which are analogous to Erdős-Gallai’s characterization using a system of inequalities. These characterizations partially answer one problem due to Lai and Hu [10].

An unoriented flow in a graph is an assignment of real numbers to the edges such that the sum of the values of all edges incident with each vertex is zero. This is equivalent to a flow in a bidirected graph where all edges are extraverted. A nowhere-zero unoriented \(k\)-flow is an unoriented flow with values from the set \(\{\pm 1, \ldots, \pm( k-1)\}\). It has been conjectured that if a graph admits a nowhere-zero unoriented flow, then it also admits a nowhere-zero unoriented \(6\)-flow. We prove that this conjecture holds true for Hamiltonian graphs, with \(6\) replaced by \(12\).

Let \(G\) be a graph with vertex set \(V(G)\), \(d_G(u,v)\) and \(\delta_G(v)\) denoteas the topological distance between vertices \(u\) and \(v\) in \(G\), and \(d_G(v)\) as the degree of vertex \(v\) in \(G\),respectively. The Schultz polynomial of \(G\) is defined as \(H^+(G) = \sum\limits_{u,v \subseteq V(G)} (\delta _G(u)+\delta _G(v))x^{d_G(u,v)}\), and the modified Schultz polynomial of \(G\) is defined as \(H^*(G) = \sum\limits_{u,v \subseteq V(G)}(\delta _G(u)+\delta _G(v)) x^{d_G(u,v)}\). In this paper, we obtain explicit analytical expressions for the expected values of the Schultz polynomial and modified Schultz polynomial of a random benzenoid chain with $n$ hexagons. Furthermore, we derive expected values of some related topological indices.

For a graph \(G\), let \(\mathcal{D}(G)\) be the set of all strong orientations of \(G\). The orientation number of \(G\), denoted by \(\vec{d}(G)\), is defined as \(\min\{d(D) \mid D \in \mathcal{D}(G)\}\), where \(d(D)\) denotes the diameter of the digraph \(D\). In this paper, we prove that \(\vec{d}(P_3 \times K_5) = 4\) and \(\vec{d}(C_8 \times K_3) = 6\), where \(\times\) is the tensor product of graphs.

In this paper, we consider the domination number, the total domination number, the restrained domination number, the total restrained domination number and the strongly connected domination number of lexicographic product digraphs.