In this paper, we give an alternative and more intuitive proof to one of two classic inequalities given by Diaconis and Graham in 1977. The inequality involves three metrics on the symmetric group, i.e., the set of all permutations of the first \( n \) positive integers. Our technique for the proof of the inequality allows us to resolve an open problem posed in that paper: When does equality hold? It also allows us to estimate how often equality holds. In addition, our technique can sometimes be applied for the proof of other inequalities between metrics or pseudo-metrics on the symmetric group.

Pooling designs are standard experimental tools in many biotechnical applications. In this paper, we construct a family of error-correcting pooling designs with the incidence matrix of two types of subspaces of singular linear space over finite fields, and exhibit their disjunct properties.

Let \( S \) be an orthogonal polygon in the plane, bounded by a simple closed curve, and let \( R \) be the smallest rectangular region containing \( S \). Assume that \( S \) is star-shaped via staircase paths. For every point \( p \) in \( \mathbb{R}^2 \setminus (\text{int} \, S) \), there is a corresponding point \( q \) in \( \text{bdry} \, S \) such that \( p \) lies in a maximal staircase convex cone \( C_q \) at \( q \) in \( \mathbb{R}^2 \setminus (\text{int} \, S) \). Furthermore, point \( q \) may be selected to satisfy these requirements:

1. If \( p \in \mathbb{R}^2 \setminus (\text{int} \, R) \), then \( q \) is an endpoint of an extreme edge of \( S \).
2. If \( p \in (\text{int} \, R) \setminus (\text{int} \, S) \), then \( q \) is a point of local nonconvexity of \( S \) and \( C_q \) is unique. Moreover, there is a neighborhood \( N \) of \( q \) such that, for \( s \) in \( (\text{bdry} \, S) \cap N \) and for \( C_s \) any staircase cone at \( s \) in \( \mathbb{R}^2 \setminus (\text{int} \, S) \), \( C_s \subseteq C_q \).

Thus we obtain a finite family of staircase convex cones whose union is \( \mathbb{R}^2 \setminus (\text{int} \, S) \).

If there are integers \( k \) and \( \lambda \neq 0 \) such that a total labeling \( f \) of a connected graph \( G = (V, E) \) from \( V \cup E \) to \( \{1, 2, \ldots, |V| + |E|\} \) satisfies \( f(x) \neq f(y) \) for distinct \( x, y \in V \cup E \) and

\[ f(u) + f(v) = k + \lambda f(uv) \]

for each edge \( uv \in E \), then \( f \) is called a \( (k, \lambda) \)-\({magically\; total\; labeling}\) (\( (k, \lambda) \)-\({mtl}\) for short) of \( G \). Several properties of \( (k, \lambda) \)-\({mtls}\) of graphs are shown. The sufficient and necessary connections between \( (k, \lambda) \)-\emph{mtls} and several known labelings (such as graceful, odd-graceful, felicitous, and \( (b, d) \)-edge antimagic total labelings) are given. Furthermore, every tree is proven to be a subgraph of a tree having super \( (k, \lambda) \)-\({mtls}\).

Let \( G \) be a simple graph of order \( n \), and let \( k \) be a positive integer. A graph \( G \) is fractional independent-set-deletable \( k \)-factor-critical (in short, fractional ID-\( k \)-factor-critical) if \( G – I \) has a fractional \( k \)-factor for every independent set \( I \) of \( G \). In this paper, we obtain a sufficient condition for a graph \( G \) to be fractional ID-\( k \)-factor-critical. Furthermore, it is shown that the result in this paper is best possible in some sense.

Calculations of the number of equivalence classes of Sudoku boards has to this point been done only with the aid of a computer, in part because of the unnecessarily large symmetry group used to form the classes. In particular, the relationship between relabeling symmetries and positional symmetries such as row/column swaps is complicated. In this paper, we focus first on the smaller Shidoku case and show first by computation and then by using connectivity properties of simple graphs that the usual symmetry group can in fact be reduced to various minimal subgroups that induce the same action. This is the first step in finding a similar reduction in the larger Sudoku case and for other variants of Sudoku.

Let \( k \) be a positive integer, and let \( G \) be a simple graph with vertex set \( V(G) \). A function \( f: V(G) \to \{\pm1, \pm2, \ldots, \pm k\} \) is called a signed \(\{k\}\)-dominating function if

\[ \sum_{u \in N[v]} f(u) \geq k \]

for each vertex \( v \in V(G) \).

The signed \(\{1\}\)-dominating function is the same as the ordinary signed domination. A set \( \{f_1, f_2, \ldots, f_d\} \) of signed \(\{k\}\)-dominating functions on \( G \) with the property that

\[ \sum_{i=1}^d f_i(v) \leq k \]

for each \( v \in V(G) \), is called a \({signed \;\{k\}-dominating \;family}\) (of functions) on \( G \). The maximum number of functions in a signed \(\{k\}\)-dominating family on \( G \) is the \({signed \;\{k\}-domatic\; number}\) of \( G \), denoted by \( d_{\{k\}S}(G) \). Note that \( d_{\{1\}S}(G) \) is the classical signed domatic number \( d_s(G) \).

In this paper, we initiate the study of signed \(\{k\}\)-domatic numbers in graphs, and we present some sharp upper bounds for \( d_{\{k\}S}(G) \). In addition, we determine \( d_{\{k\}S}(G) \) for several classes of graphs. Some of our results are extensions of known properties of the signed domatic number.

A graceful \( n \)-permutation is a graceful labeling of an \( n \)-vertex path \( P_n \). In this paper, we improve the asymptotic lower bound on the number of such permutations from \( \Omega\left(\left(\frac{5}{3}\right)^n\right) \) to \( \Omega\left(2.37^n\right) \). This is a computer-assisted proof based on an effective algorithm that enumerates graceful \( n \)-permutations. Our algorithm is also presented in detail.

Let \( K_{r+1} \) be the complete graph on \( r+1 \) vertices and let \( \pi = (d_1, d_2, \ldots, d_n) \) be a non-increasing sequence of nonnegative integers. If \( \pi \) has a realization containing \( K_{r+1} \) as a subgraph, then \( \pi \) is said to be potentially \( K_{r+1} \)-graphic. A.R. Rao obtained an Erdős-Gallai type criterion for \( \pi \) to be potentially \( K_{r+1} \)-graphic. In this paper, we provide a simplification of this Erdős-Gallai type criterion. Additionally, we present the Fulkerson-Hoffman-McAndrew type criterion and the Hasselbarth type criterion for \( \pi \) to be potentially \( K_{r+1} \)-graphic.

In this paper, we introduce several concepts related to fuzzy algebraic structures. We provide an example of a fuzzy binary operation and a fuzzy group. Additionally, we define a new fuzzy binary operation on a \(\Gamma\)-ring \(M\) and introduce a new fuzzy \(\Gamma\)-ring. We also present homomorphism theorems between two fuzzy \(\Gamma\)-rings and investigate some related properties.