Given a distribution \(D\) of pebbles on the vertices of a graph \(G\), a pebbling move on \(G\) consists of removing two pebbles from a vertex and placing one on an adjacent vertex (the other is discarded). The pebbling number of \(G\), denoted \(f(G)\), is the smallest integer \(k\) such that any distribution of \(k\) pebbles on \(G\) allows one pebble to be moved to any specified vertex via pebbling moves. In this paper, we calculate the \(t\)-pebbling number of the graph \(D_{n,C_{2m}}\). Furthermore, we verify the \(q\)-\(t\)-pebbling number to demonstrate that \(D_{n,C_{2m}}\) possesses the \(2t\)-pebbling property.

Most. of pooling designs are always constructed by the “containment matrix”. But we are interested in considering non-containment
relationship. In [J. Guo, K. Wang, Pooling designs with surprisingly high degree of error correction in a finite vector space, Discrete Appl Math], Guo and Wang gave a construction by the use of non-containment relationship. In this paper, we generalize Guo-Wang’s designs and obtain a new family of pooling designs. Our designs and Guo-Wang’s designs have the same numbers of items and pools,but the error-tolerance property of our designs is better than that of Guo-Wang’s designs.

A \(k\)-edge labeling of a graph \(G\) is a function \(f: E(G) \to \{0, \ldots, k-1\}\). Such a labeling induces a labeling on the vertex set \(V(G)\) by defining \(f(v) := \sum f(e) \pmod{k}\), where the summation is taken over all edges \(e\) incident on \(v\). For an edge labeling \(f\), let \(v_f(i)\) (resp., \(e_f(i)\)) denote the number of vertices (resp., edges) receiving the label \(i\). A graph \(G\) is said to be \(E_k\)-cordial if there exists a \(k\)-edge labeling \(f\) of \(G\)such that \(|v_f(i) – v_f(j)| \leq 1\) and \(|e_f(i) – e_f(j)| \leq 1\) for all \(0 \leq i, j \leq k-1\). A wheel \(W_n\) is the join of the cycle \(C_n\) on \(n\) vertices and \(K_1\). A Helm \(H_n\) is obtained by attaching a pendent edge to each vertex of the cycle of the wheel \(W_n\). We prove that (i) Helms, (ii) one-point unions of helms, and (iii) path unions of helms are \(E_3\)-cordial.

In this paper, we prove that the graphs \(P_n\) (\(n \geq 3\)), \(C_n\) (\(n \geq 3\), \(n \not\equiv 4 \pmod{8}\)), and \(K_n\) (\(n \geq 3\)) are \(E_4\)-cordial graphs. Additionally, we show that every graph of \(\geq 3\) is a subgraph of an \(E_4\)-cordial graph.

In this paper, we study the upper bounds for the \(D(\beta)\)-vertex-distinguishing total-chromatic numbers using the probability method, and obtain: Let \(\Delta\) be the maximum degree of \(G\), then

\[
\chi_{\beta vt}\leq
\left\{
\begin{array}{ll}
16\Delta^{(\beta+1)/(2\Delta+2)}, & \Delta \geq 3,\beta\geq 4\Delta+3; \\
13\Delta^{(\beta+4)/4} , & \Delta\geq 4,\beta\geq 5;\\
10\Delta^2, & \Delta \geq 3, 2 \leq \beta \leq 4.
\end{array}
\right.
\]

Given a tournament \(T = (V, A)\), a subset \(X\) of \(V\) is an interval of \(T\) provided that for any \(a, b \in X\) and \(x \in V \setminus X\), \((a, x) \in A\) if and only if \((b, x) \in A\). For example, \(\emptyset\), \(\{x\}\) (\(x \in V\)), and \(V\) are intervals of \(T\), called trivial intervals. A two-element interval of \(T\) is called a duo of \(T\). Tournaments that do not admit any duo are called duo-free tournaments. A vertex \(x\) of a duo-free tournament is \(d\)-critical if \(T – x\) has at least one duo. In 2005, J.F. Culus and B. Jouve [5] characterized the duo-free tournaments, all of whose vertices are d-critical, called tournaments without acyclic interval. In this paper, we characterize the duo-free tournaments that admit exactly one non-d-critical vertex, called (-1)-critically duo-free tournaments.

The toughness, as the parameter for measuring stability and vulnerability of networks, has been widely used in computer communication
networks and ontology graph structure analysis. A graph \(G\) is called a fractional \((a, b, n)\)-critical deleted graph if after deleting any \(n\) vertices from \(G\), the resulting graph is still a fractional \((a, b)\)-deleted graph. In this paper,we study the relationship between toughness and fractional \((a, b, n)\)-critical deleted graph. A sufficient condition for a graph G to be a fractional \((a, b, n)\)-critical deleted graph is determined.