In this paper, the joint tree method of graph embeddings, which was introduced by Liu, is generalized to digraph embeddings. The genus distributions of a new type of digraphs in orientable surfaces are determined.

In this paper, the \(m\)-hull sets in the join and composition of two connected graphs are characterized and their \(m\)-hull numbers are shown to be direct consequences of these characterizations.

The generalized de Bruijn digraph denoted by \(G_B(n,m)\) is the digraph \((V, A)\) where \(V = \{0,1,\ldots,m-1\}\) and \((i,j) \in A\) if and only if \(j \equiv ni + \alpha \pmod{m}\) for some \(\alpha \in \{0,1,\ldots,n-1\}\). By replacing each arc of \(G_B(n,m)\) with an undirected edge and eliminating loops and multi-edges, we obtain a generalized undirected de Bruijn graph \(UG_B(n,m)\). In this paper, we prove that the diameter of \(UG_B(n,m)\) is equal to 3 whenever \(n \geq 2\) and \(n^2 + (\frac{\sqrt{5}+1}{2})\leq m \leq 2n^2.\)

The zeroth-order general Randić index of a graph \(G\) is defined as \({}^{0}{}{R}_\alpha = \sum\limits_{v\in V(G)} d(v)^\alpha\)
where \(d(v)\) is the degree of the vertex \(v\) in \(G\) and \(\alpha\) is an arbitrary real number. In the paper, we give sharp lower and upper bounds on the zeroth-order general Randić index of cacti.

Let \(G\) be a connected \(k\)-colourable graph of order \(n \geq k\). A subgraph \(H\) of \(G\) is \(k\)-colourfully panconnected in \(G\) if there is a \(k\)-colouring of \(G\) such that the colours are close together in \(H\), in two different senses (called variegated and panconnected) to be made precise. Let \(s_k(G)\) denote the smallest number of edges in a spanning \(k\)-colourfully panconnected subgraph \(H\) of \(G\). It is conjectured that \(s_k(G) = n-1\) if \(k \geq 4\) and \(G\) is not a circuit (a connected \(2\)-regular graph) with length \(\equiv 1 \pmod{k}\). It is proved that \(s_k(G) = n-1\) if \(G\) contains no circuit with length \(\equiv 1 \pmod{k}\), and \(s_k(G) \leq 2n-k-1\) whenever \(k \geq 4\).

Multisender authentication codes allow a group of senders to construct an authenticated message for a receiver such that the receiver can verify authenticity of the received message. In this paper, we give the model of multisender authentication codes and the calculation formulas on probability of success in attacks by malicious groups of senders. A construction of multisender authentication codes from symplectic geometry over finite fields is given, and the parameters and the probabilities of deceptions are also calculated.

Let \((X,{B})\) be an \(\alpha\)-fold block design with block size \(4\). If a star is removed from each block of \({B}\), the resulting collection of triangles \({T}\) is a partial \(\lambda\)-fold triple system \((X,{T})\). If the edges belonging to the deleted stars can be arranged into a collection of triangles \({S}^*\), then \((X,{T} \cup {S}^*)\) is an \(\lambda\)-fold triple system, called a metamorphosis of the \(\lambda\)-fold block design \((X, {B})\) into a \(4\)-fold triple system.

Label the elements of each block \(b\) with \(b_1, b_2, b_3\) and \(b_4\) (in any manner). For each \(i = 1,2,3,4\), define a set of triangles \({T}_i\) and a set of stars \({S}_i\) as follows: for each block \(b = (b_1, b_2, b_3, b_4)\) belonging to \({B}\), partition \(b\) into a triangle and a star centered at \(b_i\), and place the triangle in \({T}_i\) and the star in \({S}_i\). Then \((X,\mathcal{T}_i)\) is a partial \(\alpha\)-fold triple system.

Now if the edges belonging to the stars in \({S}_i\) can be arranged into a collection of triangles \({S}_i^*\), then \((X,{T}_i \cup {S}_i^*)\) is an \(\lambda\)-fold triple system and we say that \(M_i = (X,{T}_i \cup {S}_i^*)\) is the \(i\)th metamorphosis of \((X,{B})\).

The full metamorphosis of \((X,{B})\) is the set of four metamorphoses \(\{M_1, M_2, M_3, M_4\}\). The purpose of this work is to give a complete solution of the following problem: For which \(n\) and \(\lambda\) does there exist an \(\lambda\)-fold block design with block size \(4\) having a full metamorphosis into \(\lambda\)-fold triple systems?

A labeling of a graph is any map that carries some set of graph elements to numbers (usually to the positive integers). An \((a, d)\)-edge-antimagic total labeling on a graph with \(p\) vertices and \(q\) edges is defined as a one-to-one map taking the vertices and edges onto the integers \(1,2,…,p+q\) with the property that the sums of the labels on the edges and the labels of their endpoints form an arithmetic sequence starting from \(a\) and having a common difference \(d\). Such a labeling is called super if the smallest possible labels appear on the vertices.

We use the connection between \(a\)-labelings and edge-antimagic labelings for determining a super \((a,d)\)-edge-antimagic total labelings of disconnected graphs.

Let \(P\) be an \(n \times n\) array of symbols. \(P\) is called avoidable if for every set of \(z\) symbols, there is an \(n \times n\) Latin square \(L\) on these symbols so that corresponding cells in \(P\) and \(L\) differ. Due to recent work of Cavenagh and Ohman, we now know that all \(n \times n\) partial Latin squares are avoidable for \(n \geq 4\). Cavenagh and Ohman have shown that partial Latin squares of order \(4m + 1\) for \(m \geq 1\) [1] and \(4m – 1\) for \(m \geq 2\) [2] are avoidable. We give a short argument that includes all partial Latin squares of these orders of at least \(9\). We then ask the following question: given an \(n \times n\) partial Latin square \(P\) with some specified structure, is there an \(n \times n\) Latin square \(L\) of the same structure for which \(L\) avoids \(P\)? We answer this question in the context of generalized sudoku squares.

For a graph \(G\) with vertices labeled \(1,2,\ldots,n\) and a permutation \(\alpha\) in \(S_n\), the symmetric group on \(\{1,2,\ldots,n\}\), the \(\alpha\)-generalized prism over \(G\), \(\alpha(G)\), consists of two copies of \(G\), say \(G_x\) and \(G_y\), along with the edges \((x_i, y_{\alpha(i)})\), for \(1 \leq i \leq n\). In [10], the importance of building large graphs by using generalized prisms is indicated. A graph \(G\) is supereulerian if it has a spanning eulerian subgraph. In this note, we consider results of the form that if \(G\) has property \(P\), then for any \(\alpha \in S_{|V(G)|}\), \(\alpha(G)\) is supereulerian. As a result, we obtain a few properties of \(G\) which implies that for any \(\alpha \in S_{|V(G)|}\), \(\alpha(G)\) is supereulerian. Also, while the permutations are restricted, the related result is discussed.