We pursue the problem of counting the imbeddings of a graph in each of the orientable surfaces. We demonstrate how to achieve this for an iterated amalgamation of arbitrarily many copies of any graph whose genus distribution is known and further analyzed into a partitioned genus distribution. We introduce the concept of recombinant strands of face-boundary walks, and we develop the use of multiple production rules for deriving simultaneous recurrences. These two ideas are central to a broad-based approach to calculating genus distributions for graphs synthesized from smaller graphs.

The super (resp., edge-) connectivity of a connected graph is the minimum cardinality of a vertex-cut (resp., an edge-cut) whose removal does not isolate a vertex. In this paper, we consider the two parameters for a special class of graphs \(G(G_p,G_1; M)\), proposed by Chen et al [Applied Math. and Computation, \(140 (2003), 245-254]\), obtained from two \(k\)-regular \(k\)-connected graphs \(G_p\) and \(G_1\), with the same order by adding a perfect matching between their vertices. Our results improve ones of Chen et al. As applications, the super connectivity and the super edge-connectivity of the \(n\)-dimensional hypercube, twisted cube, cross cube, Möbius cube and locally twisted cube are all \(2n – 2\).

We investigate the existence of \(3\)-designs and uniform large sets of \(3\)-designs with block size \(6\) admitting \(\text{PSL}(2, 2^n)\) as an automorphism group.

In \([5]\), a product summation of ordered partition \(f(n,m,r) = \sum{c_1^r + c_2^r + \cdots + c_m^r }\) was defined, where for two given positive integers \(m,r\), the sum is over all positive integers \(c_1, c_2, \ldots, c_m\) with \(c_1 + c_2 + \cdots + c_m = n\). \(f(n,r) = \sum_{i=1}^n f(n,m,r)\) was also defined. Many results on \(f(n,m,r)\) were found. However, few things have been known about \(f(n,r)\). In this paper, we give more details for \(f(n,r)\), including its two recurrences, its explicit formula via an entry of a matrix and its generating function. Unexpectedly, we obtain some interesting combinatorial identities, too.

In this paper, we obtain new general results containing sums of binomial and multinomial with coefficients satisfying a general third order linear recursive relations with indices in arithmetic progression.

The closed neighborhood \(N[e]\) of an edge \(e\) in a graph \(G\) is the set consisting of \(e\) and of all edges having a common end-vertex with \(e\). Let \(f\) be a function on \(E(G)\), the edge set of \(G\), into the set \(\{-1,1\}\). If \(\sum_{e \in N[e]} f(x) \geq 1\) for each \(e \in E(G)\), then \(f\) is called a signed edge dominating function of \(G\). The minimum of the values \(\sum_{e \in E(G)} f(e)\), taken over all signed edge dominating functions \(f\) of \(G\), is called the signed edge domination number of \(G\) and is denoted by \(\gamma’_s(G)\). It has been conjectured that \(\gamma’_s(T) \geq 1\) for every tree \(T\). In this paper we prove that this conjecture is true and then classify all trees \(T\) with \(\gamma’_s(T) = 1,2\) and \(3\).

This article is a contribution to the study of block-transitive automorphism groups of \(2\)-\((v,k,1)\) block designs. Let \(\mathcal{D}\) be a \(2\)-\((v,k,1)\) design admitting a block-transitive, point-primitive but not flag-transitive group \(G\) of automorphisms. Let \(k_1 = (k, v-1)\) and \(q = p^f\) for prime \(p\). In this paper we prove that if \(G\) and \(D\) are as above and \(q > {(2(k_rk-k_r+1)f)^{\frac{1}{4}}}\) then \(G\) does not admit a Chevalley group \(E_7(q)\) as its socle.

A graph \(G\) is called super edge-magic if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that \(f(u) + f(v) + f(uv) = C\) is a constant for any \(uv \in E(G)\) and \(f(V(G)) = \{1, 2, \ldots, |V(G)|\}\), \(f(E(G)) = \{|V(G)| + 1, |V(G)| + 2, \ldots, |V(G)| + |E(G)|\}\). R. M. Figueroa-Centeno et al. provided the following conjecture: For every integer \(n \geq 5\), the book \(B_n\) is super edge-magic if and only if \(n\) is even or \(n \equiv 5 \pmod 8\). In this paper, we show that \(B_n\) is super edge-magic for even \(n \geq 6\).

It was conjectured in \([10]\) that the upper bound for the strong chromatic index \(s'(G)\) of bipartite graphs is \(\Delta(G)^2+1\), where \(\Delta(G)\) is the largest degree of vertices in \(G\). In this note we study the strong edge coloring of some classes of bipartite graphs that belong to the class of partial cubes. We introduce the concept of \(\Theta\)-graph \(\Theta(G)\) of a partial cube \(G\), and show that \(s'(G) \leq \chi(\Theta(G))\) for every tree-like partial cube \(G\). As an application of this bound we derive that \(s'(G) \leq 2\Delta(G)\) if \(G\) is a \(p\)-expansion graph.

We introduce notions of \(k\)-chromatic uniqueness and \(k\)-chromatic equivalence in the class of all Sperner hypergraphs. They generalize the chromatic uniqueness and equivalence defined in the class of all graphs \([10]\) and hypergraphs \([2, 4, 8]\). Using some known facts, concerning a \(k\)-chromatic polynomial of a hypergraph \([5]\), a set of hypergraphs whose elements are \(3\)-chromatically unique is indicated. A set of hypergraphs characterized by a described \(3\)-chromatic polynomial is also shown. The application of the investigated notions can be found in \([5]\).