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.

Using the model of words, we give bijective proofs of Gould-Mohanty’s and Raney-Mohanty’s identities, which are respectively multivariable generalizations of Gould’s identity

\[\sum\limits_{k=0}^{n} \left(
\begin{array}{c}
x-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y+kz \\
n-k \\
\end{array}
\right)
= \sum\limits_{k=0}^{n}
\left(
\begin{array}{c}
x+\epsilon-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y-\epsilon+kz \\
n-k \\
\end{array}
\right)
\]

and Rothe’s identity
\[\sum\limits_{k=0}^{n}\frac{x}{x-kz}
\left(
\begin{array}{c}
x-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y+kz \\
n-k \\
\end{array}
\right)
=
\left(
\begin{array}{c}
x+y \\
n \\
\end{array}
\right)\]

Ryjáček introduced a closure concept in claw-free graphs based on local completion at a locally connected vertex. He showed that the closure of a graph is the line graph of a triangle-free graph. Broušek and Holub gave an analogous closure concept of claw-free graphs, called the edge-closure, based on local completion at a locally connected edge. In this paper, it is shown that the edge-closure is the line graph of a multigraph.

For a graph \(G = (V,E)\), a function \(f : V \rightarrow \{0,1,2\}\) is called a Roman dominating function (RDF) if for any vertex \(v\) with \(f(v) = 0\), there is at least one vertex \(w\) in its neighborhood with \(f(w) = 2\).

The weight of an RDF \(f\) of \(G\) is the value \(f(V) = \sum_{v\in V} f(v)\). The minimum weight of an RDF of \(G\) is its Roman domination number, denoted by \(\gamma_R(G)\). In this paper, we show that \(\gamma_R(G) + 1 \leq \gamma_R(\mu(G)) \leq \gamma_R(G) + 2\), where \(\mu(G)\) is the Mycielekian graph of \(G\), and then characterize the graphs achieving equality in these bounds.

A graph is said to be cordial if it has a \(0-1\) labeling that satisfies certain properties. A fan \(F_n\) is the graph obtained from the join of the path \(P_n\) and the null graph \(N_1\). In this paper, we investigate the cordiality of the join and the union of pairs of fans and graphs consisting of a fan with a path, and a cycle.

We consider the problem of covering a unit cube with smaller cubes. The size of a cube is given by its side length and the size of a covering is the total size of the cubes used to cover the unit cube. We denote by \(g_3(n)\) the smallest size of a minimal covering using \(n\) cubes. We present tight results for the upper and lower bounds of \(g_3(n)\).

Let \(G\) be a graph. The cardinality of any largest independent set of vertices in \(G\) is called the independence number of \(G\) and is denoted by \(\alpha(G)\). Let \(a\) and \(b\) be integers with \(0 \leq a \leq b\). If \(a = b\), it is assumed that \(G\) be a connected graph, furthermore, \(a \geq \alpha(G)\), \(a/|V(G)| = 0 \pmod{2}\) if \(a\) is odd. We prove that every graph \(G\) has an \([a, b]\)-factor if its minimum degree is at least \((\frac{b+\alpha(G)a-\alpha(G)}{b})\lfloor \frac{b+\alpha(G)a}{2\alpha(G)} \rfloor -\frac{\alpha(G)}{b}(\lfloor \frac{b+\alpha(G)a}{2\alpha(G)}\rfloor )^2+ \theta\frac{\alpha(G)^2}{b}+\frac{a}{b}\alpha(G)\), where \(\theta = 0\) if \(a < b\), and \(\theta = 1\) if \(a = b\). This degree condition is sharp.