A perfect code in a graph Γ = (V,E) is a subset C of V that is an independent set such that every vertex in V \ C is adjacent to exactly one vertex in C. A total perfect code in Γ is a subset C of V such that every vertex of V is adjacent to exactly one vertex in C. A perfect code in the Hamming graph H(n, q) agrees with a q-ary perfect 1-code of length n in the classical setting. A necessary a...

We prove that optimal strategies exist in perfect-information stochastic games with finitely many states and actions and tail winning conditions. Introduction We prove that optimal strategies exist in perfect-information stochastic games with finitely many states and actions and tail winning conditions. This proof is different from the algorithmic proof sketched in [Hor08]. 1. Perfect-Informati...

Let G be a group. Two elements x, y are said to be in the same z-class if their centralizers are conjugate in G. The conjugacy classes give a partition of G. Further decomposition of the conjugacy classes into z-classes provides an important information about the internal structure of the group. Let V be a vector space of dimension n over a field F of characteristic different from 2. Let B be a...

In a Steiner triple system STS(v)=(V,B), for each pair {a, b} ⊂ V , the cycle graph Ga,b can be defined as follows. The vertices of Ga,b are V \{a, b, c} where {a, b, c} ∈ B. {x, y} is an edge if either {a, x, y} or {b, x, y} ∈ B. The Steiner triple system is said to be perfect if the cycle graph of every pair is a single (v − 3)-cycle. Perfect STS(v) are known only for v = 7, 9, 25 and 33. We ...

Let G = (V,E) be a graph and let S & V. The set S is a dominating set of G is every vertex of V-S is adjacent to a vertex of S. A vertex v of G is called S-perfect if \N[t~]nsi = 1 where N[v] denotes the closed neighborhood of v. The set S is defined to be a perfect neighborhood set of G if every vertex of G is S-perfect or adjacent with an S-perfect vertex. We prove that for all graphs G, O(G)...

A k−geodominating set is a geodominating set S such that any vertex v ∈ V (G)\S is geodominated by a pair x, y of vertices of S with d(x, y) = k. A k-perfect geodominating set is a geodominating set S such that any vertex v ∈ V (G) \ S is geodominated by exactly one pair x, y of the vertices of S with d(x, y) = k. The cardinality of a minimum perfect geodominating set in G is its perfect geodom...

Let k be a perfect field of virtual cohomological dimension ≤ 2. Let G be a connected linear algebraic group over k such that Gsc satisfies a Hasse principle over k. Let X be a principal homogeneous space under G over k. We show that if X admits a zero cycle of degree one, then X has a k-rational point. Introduction The following question of Serre [10, p. 192] is open in general. Q: Let k be a ...

Given an undirected network G(V,A,c) and a perfect matching M of G, the inverse maximum perfect matching problem consists of modifying minimally the elements of c so that M becomes a maximum perfect matching with respect to the modified vector. In this article, we consider the inverse problem when the modifications are measured by the weighted bottleneck-type Hamming distance. We propose an alg...

A restricted permutation of a locally finite directed graph $ G = (V, E) is vertex \pi: V\to V for which (v, \pi(v))\in E $, any v\in $. The set such permutations, denoted by \Omega(G) with group action induced from subset isomorphisms form topological dynamical system. We focus on the particular case presented Schmidt and Strasser [18] {\mathbb Z}^d in subshift type. show correspondence betwee...