For an \(n\)-connected graph \(G\), the \(n\)-wide diameter \(d_n(G)\), is the minimum integer \(m\) such that there are at least \(n\) internally disjoint \((di)\)paths of length at most \(m\) between any vertices \(x\) and \(y\). For a given integer \(l\), a subset \(S\) of \(V(G)\) is called an \((l, n)\)-dominating set of \(G\) if for any vertex \(x \in V(G) – S\) there are at least \(n\) internally disjoint \((di)\)paths of length at most \(l\) from \(S\) to \(z\). The minimum cardinality among all \((l, n)\)-dominating sets of \(G\) is called the \((l, n)\)-domination number. In this paper, we obtain that the \((l, n)\)-domination number of the \(d\)-ary cube network \(C(d, n)\) is \(2\) for \(1 \leq w \leq d\) and \(d_w(G) – f(d, n) \leq l \leq d_w(G) – 1 \) if \(d,n\geq 4\), where \(f(d, n) = \min\{e(\left\lfloor \frac{n}{2} \right\rceil + 1), \left\lfloor \frac{n}{2} \right\rfloor e\}\).

Let \(k\) be a positive integer, and let \(G\) be a simple graph with vertex set \(V(G)\). A function \(f: V(G) \rightarrow \{-1, 1\}\) is called a signed \(k\)-dominating function if \(\sum_{u \in N(v)} f(u) \geq k\) for each vertex \(v \in V(G)\). A set \(\{f_1, f_2, \ldots, f_d\}\) of signed \(k\)-dominating functions on \(G\) with the property that \(\sum_{i=1}^{d} f_i(v) \leq 1\) for each \(v \in V(G)\), is called a signed \(k\)-dominating family (of functions) on \(G\). The maximum number of functions in a signed \(k\)-dominating family on \(G\) is the signed \(k\)-domatic number of \(G\), denoted by \(d_{kS}(G)\). In this paper, we initiate the study of signed \(k\)-domatic numbers in graphs and we present some sharp upper bounds for \(d_{kS}(G)\). In addition, we determine the signed \(k\)-domatic number of complete graphs.

In this paper, \(E_2\)-cordiality of a graph \(G\) is considered. Suppose \(G\) contains no isolated vertex, it is shown that \(G\) is \(E_2\)-cordial if and only if \(G\) is not of order \(4n + 2\).

A Hamiltonian walk of a connected graph \(G\) is a closed spanning walk of minimum length in \(G\). The length of a Hamiltonian walk in \(G\) is called the Hamiltonian number, denoted by \(h(G)\). An Eulerian walk of a connected graph \(G\) is a closed walk of minimum length which contains all edges of \(G\). In this paper, we improve some results in [5] and give a necessary and sufficient condition for \(h(G) < e(G)\). Then we prove that if two nonadjacent vertices \(u\) and \(v\) satisfying that \(\deg(u) + \deg(v) \geq |V(G)|\), then \(h(G) = h(G + uv)\). This result generalizes a theorem of Bondy and Chvatal for the Hamiltonian property. Finally, we show that if \(0 \leq k \leq n-2\) and \(G\) is a 2-connected graph of order \(n\) satisfying \(\deg(u) + \deg(v) + \deg(w) \geq \frac{3n+k-2}{2}\) for every independent set \(\{u,v,w\}\) of three vertices in \(G\), then \(h(G) \leq n+k\). It is a generalization of Bondy's result.

Let \(G\) be a finite abelian group of order \(n\). The barycentric Ramsey number \(BR(H,G)\) is the minimum positive integer \(r\) such that any coloring of the edges of the complete graph \(K_r\) by elements of \(G\) contains a subgraph \(H\) whose assigned edge colors constitute a barycentric sequence, i.e., there exists one edge whose color is the “average” of the colors of all edges in \(H\). When the number of edges \(e(H) \equiv 0 \pmod{\exp(G)}\), \(BR(H,G)\) are the well-known zero-sum Ramsey numbers \(R(H,G)\). In this work, these Ramsey numbers are determined for some graphs, in particular, for graphs with five edges without isolated vertices using \(G = \mathbb{Z}_n\), where \(2 \leq n \leq 4\), and for some graphs \(H\) with \(e(H) \equiv 0 \pmod{2}\) using \(G = \mathbb{Z}_2^s\).