Let \(G\) be a finite group and \(S\) be a subset (possibly containing the identity element) of \(G\). We define the Bi-Cayley graph \(X = BC(G, S)\) to be the bipartite graph with vertices \(G \times \{0, 1\}\) and edges \(\{(g, 0), (sg, 1) : g \in G, s \in S\}\). In this paper, we show that if \(X = BC(G, S)\) is connected, then \(\kappa(X) = \delta(X)\).

Some new characterizations for harmonic Bergman space on the unit ball \({B}\) in \(\mathbb{R}^n\) are given in this paper. They can be described as derivative-free characterizations.

The planar Ramsey number \(PR(H_1, H_2)\) is the smallest integer \(n\) such that any planar graph on \(n\) vertices contains a copy of \(H_1\) or its complement contains a copy of \(H_2\). It is known that the Ramsey number \(R(K_4 – e, K_k – e)\) for \(k \leq 6\). In this paper, we prove that \(PR(K_4 – e, K_6 – e) = 16\) and show the lower bounds on \(PR(K_4 – e, K_k – e)\).

Let \(K_v\) be a complete graph with \(v\) vertices, and \(G = (V(G), E(G))\) be a finite simple graph. A \(G\)-design \(G-GD_\lambda(v)\) is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in exactly \(\lambda\) blocks of \(\mathcal{B}\). In this paper, the existence of graph designs \(G-GD_\lambda(v)\), \(\lambda > 1\), for eight graphs \(G\) with six vertices and eight edges is completely solved.

A \({weighted \;graph}\) is one in which every edge \(e\) is assigned a nonnegative number \(w(e)\), called the \({weight}\) of \(e\). The \({weight\; of \;a \;cycle}\) is defined as the sum of the weights of its edges. The \({weighted \;degree}\) of a vertex is the sum of the weights of the edges incident with it. In this paper, motivated by a recent result of Fujisawa, we prove that a \(2\)-connected weighted graph \(G\) contains either a Hamilton cycle or a cycle of weight at least \(2m/3\) if it satisfies the following conditions:
\((1)\) The weighted degree sum of every three pairwise nonadjacent vertices is at least \(m\);\((2)\)In each induced claw and each induced modified claw of \(G\), all edges have the same weight.This extends a theorem of Zhang, Broersma and Li.

The \({restricted edge-connectivity}\) of a graph is an important parameter to measure fault-tolerance of interconnection networks. This paper determines that the restricted edge-connectivity of the de Bruijn digraph \(B(d,n)\) is equal to \(2d – 2\) for \(d \geq 2\) and \(n \geq 2\) except \(B(2,2)\). As consequences, the super edge-connectedness of \(B(d,n)\) is obtained immediately.

An edge coloring of a graph is called \({square-free}\) if the sequence of colors on certain walks is not a square, that is not of the form \(x_1, \ldots, x_m, x_{1}, \ldots, x_m\) for any \(m \in \mathbb{N}\). Recently, various classes of walks have been suggested to be considered in the above definition. We construct graphs, for which the minimum number of colors needed for a square-free coloring is different if the considered set of walks vary, solving a problem posed by Brešar and Klavžar. We also prove the following: if an edge coloring of \(G\) is not square-free (even in the most general sense), then the length of the shortest square walk is at most \(8|E(G)|^2\). Hence, the necessary number of colors for a square-free coloring is algorithmically computable.

If \(x\) is a vertex of a digraph \(D\), then we denote by \(d^+ (x)\) and \(d^- (x)\) the outdegree and the indegree of \(x\), respectively. The global irregularity of a digraph \(D\) is defined by \(i_g(D) = \max\{d^+ (x),d^- (x)\} – \min\{d^+ (y), d^- (y)\}\) over all vertices \(x\) and \(y\) of \(D\) (including \(x = y\)).

A \(c\)-partite tournament is an orientation of a complete \(c\)-partite graph. Recently, Volkmann and Winzen \([9]\) proved that \(c\)-partite tournaments with \(i_g(D) = 1\) and \(c \geq 3\) or \(i_g(D) = 2\) and \(c \geq 5\) contain a Hamiltonian path. Furthermore, they showed that these bounds are best possible.

Now, it is a natural question to generalize this problem by asking for the minimal value \(g(i,k)\) with \(i,k \geq 1\) arbitrary such that all \(c\)-partite tournaments \(D\) with \(i_g(D) \leq i\) and \(c \geq g(i,k)\) have a path covering number \(pc(D) \leq k\). In this paper, we will prove that \(4i-4k \leq g(i,k) \leq 4i-3k-1\), when \(i \geq k+2\). Especially in the case \(k = 1\), this yields that \(g(i, 1) = 4i-4\), which means that all \(c\)-partite tournaments \(D\) with the global irregularity \(i_g(D) = i\) and \(c \geq 4i-4\) contain a Hamiltonian path.

In this paper, we discuss a problem on packing a unit cube with smaller cubes, which is a generalization of one of Erdős’ favorite problems: the square-packing problem. We first give the definition of the packing function \(f_3(n)\), then give the bounds for \(f_3(n)\).

A set \(S\) of vertices in a graph \(G = (V, E)\) is a restrained dominating set of \(G\) if every vertex not in \(S\) is adjacent to a vertex in \(S\) and to a vertex in \(V \setminus S\). The graph \(G\) is called restrained domination excellent if every vertex belongs to some minimum restrained dominating set of \(G\). We provide a characterization of restrained domination excellent trees.