For a given graph \(H\), a graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) is said to be potentially \(H\)-graphic if there is a realization of \(\pi\) containing \(H\) as a subgraph. In this paper, we characterize potentially \(K_{1,1,6}\)-positive graphic sequences. This characterization implies the value of \(\sigma(K_{1,1,6}, n)\). Moreover, we also give a simple sufficient condition for a positive graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) to be potentially \(K_{1,1,s}\)-graphic for \(n \geq s+2\) and \(s \geq 2\).

Let \(H(B)\) denote the space of all holomorphic functions on the unit ball \(B\). Let \(u \in H(B)\) and \(\varphi\) be a holomorphic self-map of \(B\). This paper characterizes the boundedness and compactness of the weighted composition operator \(uC_{\varphi}\), from Bloch-type spaces to weighted-type spaces in the unit ball.

Let \(G\) be a graph of order \(n\). Let \(a\) and \(b\) be integers with \(1 \leq a < b\), and let \(k \geq 2\) be a positive integer not larger than the independence number of \(G\). Let \(g(x)\) and \(f(x)\) be two non-negative integer-valued functions defined on \(V(G)\) such that \(a \leq g(x) \frac{(a+b)(k(a+b)-2)}{a+1}\) and \(|N_G(x_1) \cup N_G(x_2) \cup \cdots \cup N_G(x_k)| \geq \frac{(b-1)n}{a+b}\) for any independent subset \(\{x_1, x_2, \ldots, x_k\}\) of \(V(G)\). Furthermore, we show that the result is best possible in some sense.

A graph \(G\) is called quasi-claw-free if it satisfies the property:\(d(x,y) = 2 \Rightarrow \text{there exists} u \in N(x) \cap N(y) \text{ such that } N[u] \subseteq N[x] \cup N[y].\) It is shown that a Hamiltonian cycle can be found in polynomial time in four subfamilies of quasi-claw-free graphs.

We study near hexagons which satisfy the following properties:(i) every two points at distance 2 from each other are contained in a unique quad of order \((s,r_1)\) or \((s,r_2), r_1\neq r_2\); (ii) every line is contained in the same number of quads; (iii) every two opposite points are connected by the same number of geodesics. We show that there exists an association scheme on the point set of such a near hexagon and calculate the intersection numbers. We also show how the eigenvalues of the collinearity matrix and their corresponding multiplicities can be calculated. The fact that all multiplicities and intersection numbers are nonnegative integers gives restrictions on the parameters of the near hexagon. We apply this to the special case in which the near hexagon has big quads.

A perfect \(r\)-code in a graph is a subset of the graph’s vertices with the property that each vertex in the graph is within distance \(r\) of exactly one vertex in the subset. We determine the relationship between perfect \(r\)-codes in the lexicographic product of two simple graphs and perfect \(r\)-codes in each of the factors.

A graph \(G\) is called uniquely \(k\)-list colorable, or \(UkLC\) for short, if it admits a \(k\)-list assignment \(L\) such that \(G\) has a unique \(L\)-coloring. A graph \(G\) is said to have the property \(M(k)\) (\(M\) for Marshal Hall) if and only if it is not \(UkLC\). The \(m\)-number of a graph \(G\), denoted by \(m(G)\), is defined to be the least integer \(k\) such that \(G\) has the property \(M(k)\). After M. Mahdian and E.S. Mahmoodian characterized the \(U2LC\) graphs, M. Ghebleh and E.S. Mahmoodian characterized the \(U3LC\) graphs for complete multipartite graphs except for nine graphs in 2001. Recently, W. He et al. verified all the nine graphs are not \(U3LC\) graphs. Namely, the \(U3LC\) complete multipartite graphs are completely characterized. In this paper, complete multipartite graphs whose \(m\)-number are equal to \(4\) are researched and the \(U4LC\) complete multipartite graphs, which have at least \(6\) parts, are characterized except for finitely many of them. At the same time, we give some results about some complete multipartite graphs whose number of parts is smaller than \(6\).

A list-assignment \(L\) to the vertices of \(G\) is an assignment of a set \(L(v)\) of colors to vertex \(v\) for every \(v \in V(G)\). An \((L,d)^*\)-coloring is a mapping \(\phi\) that assigns a color \(\phi(v) \in L(v)\) to each vertex \(v \in V(G)\) such that at most \(d\) neighbors of \(v\) receive color \(\phi(v)\). A graph is called \((k,d)^*\)-choosable, if \(G\) admits an \((L,d)^*\)-coloring for every list assignment \(L\) with \(|L(v)| \geq k\) for all \(v \in V(G)\). In this note, it is proved that:(1) every toroidal graph containing neither adjacent \(3\)-cycles nor \(5\)-cycles, is \((3,2)^*\)-choosable;(2) every toroidal graph without \(3\)-cycles, is \((3,2)^*\)-choosable.

In this note, we consider a generalized Fibonacci sequence \(\{u_n\}\). Then we give a generating matrix for the terms of sequence \(\{u_{kn}\}\) for a positive integer \(k\). With the aid of this matrix, we derive some new combinatorial identities for the sequence \(\{u_{kn}\}\).

Let \(G = (V, E)\) be a graph. A subset \(S\) of \(V\) is called a dominating set of \(G\) if every vertex in \(V – S\) is adjacent to at least one vertex in \(S\). A global dominating set is a subset \(S\) of \(V\) which is a dominating set of both \(G\) as well as its complement \(\overline{G}\). The domination number (global domination number) \(\gamma(\gamma_g)\) of \(G\) is the minimum cardinality of a dominating set (global dominating set) of \(G\). In this paper, we obtain a characterization of bipartite graphs with \(\gamma_g = \gamma + 1\). We also characterize unicyclic graphs and bipartite graphs with \(\gamma_g = \alpha_0(G) + 1\), where \(\alpha_0(G)\) is the vertex covering number of \(G\).