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 \(G\).

In paper \([7]\), S. J. Xu and W. Jin proved that a cyclic group of order \(pq\), for two different odd primes \(p\) and \(q\), is a \(3\)-BCI-group, and a finite \(p\)-group is a weak \((p – 1)\)-BCI-group. As a continuation of their works, in this paper, we prove that a cyclic group of order \(2p\) is a \(3\)-BCI-group, and a finite \(p\)-group is a \((p – 1)\)-BCI-group.

Fifty-five new or improved lower bounds for \(A(n, d, w)\), the maximum possible number of binary vectors of length \(n\), weight \(w\), and pairwise Hamming distance no less than \(d\), are presented.

We give some estimates of the norm of weighted composition operators from \(\alpha\)-Bloch spaces to Bloch-type spaces on the unit ball in \(7\).

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). The isolated toughness of \(G\) is defined as
\(I(G) = \min\left\{\frac{|S|}{i(G-S)}: S \subseteq V(G), i(G-S) \geq 2\right\}\)
if \(G\) is not complete. Otherwise, set \(I(G) = |V(G)| – 1\). Let \(a\) and \(b\) be positive integers such that \(1 \leq a \leq b\), and let \(g(x)\) and \(f(x)\) be positive integral-valued functions defined on \(V(G)\) such that \(a \leq g(x) \leq f(x) \leq b\). Let \(h(e) \in [0,1]\) be a function defined on \(E(G)\), and let \(d(x) = \sum_{e \in E_x} h(e)\) where \(E_x = \{xy : y \in V(G)\}\). Then \(d(x)\) is called the fractional degree of \(x\) in \(G\). We call \(h\) an indicator function if \(g(x) \leq d(x) \leq f(x)\) holds for each \(x \in V(G)\). Let \(E^h = \{e : e \in E(G), h(e) \neq 0\}\) and let \(G_h\) be a spanning subgraph of \(G\) such that \(E(G_h) = E^h\). We call \(G_h\) a fractional \((g,f)\)-factor. The main results in this paper are to present some sufficient conditions about isolated toughness for the existence of fractional \((g,f)\)-factors. If \(1 = g(x) < f(x) = b\), this condition can be improved and the improved bound is not only sharp but also a necessary and sufficient condition for a graph to have a fractional \([1,b]\)-factor.