Given a graphG = (V,E) and a set Lv of admissible colors for each vertex v ∈ V (termed the list at v), a list coloring of G is a (proper) vertex coloring φ : V → v∈V Lv such that φ(v) ∈ Lv for all v ∈ V and φ(u) 6= φ(v) for all uv ∈ E. If such a φ exists, G is said to be list colorable. The choice number of G is the smallest natural number k for which G is list colorable whenever each list cont...

Let G be a finite group, k a perfect field, and V a finite dimensional kG-module. We let G act on the power series k[[V ]] by linear substitutions and address the question of when the invariant power series k[[V ]] form a unique factorization domain. We prove that for a permutation module for a p-group in characteristic p, the answer is always positive. On the other hand, if G is a cyclic group...

Let k be a perfect field and let K/k be a finite extension of fields. An arithmetic noncommutative projective line is a noncommutative space of the form ProjSK(V ), where V be a k-central two-sided vector space over K of rank two and SK(V ) is the noncommutative symmetric algebra generated by V over K defined by M. Van den Bergh [26]. We study the geometry of these spaces. More precisely, we pr...

1.1. Let V be a complete discrete valuation ring with perfect residue field of characteristic p > 0 and fraction field K of characteristic 0. Fix an algebraic closure K ⊂ K. Let X/V be a smooth proper scheme, and let D ⊂ X be a divisor with normal crossings over V . Assume that XK is connected. Let X ◦ denote X −D and assume given a point x ∈ X◦(V ). We can then consider the pro-unipotent funda...

In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs. Given a simple graph G = (V,E) with |V | = 2n vertices such that E contains a perfect matching (of size n), together with a color (or label) function L : E → {c1, . . . , cq}, the labeled perfect matching problem consists in finding a perfect matching on G that us...

Fix a non-negative integer g and a positive integer I dividing 2g − 2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C/K of genus g and index I. This is obtained via a systematic analysis of local points on arithmetic surfaces with semistable reduction. Applications are discussed to the corresponding prob...

The Wonderful Lemma, that was first proved by Roussel and Rubio, is one of the most important tools in the proof of the Strong Perfect Graph Theorem. Here we give a short proof of this lemma.

A perfect (t, w, v; m)-threshold scheme is a type of combinatorial design that provides a way of distributing partial information (chosen from a set of v points called shadows) to w participants, so that any t of them can easily calculate one of m possible keys, but no subset of fewer than t participants can determine any partial information regarding the key. In this paper, we give a survey of...

A perfect code in a graph ?=(V,E) is subset C of V such that no two vertices are adjacent and every vertex V?C to exactly one C. subgroup H group G called if there exists Cayley which admits as code. Equivalently, an inverse-closed containing the identity element (A,H) tiling sense can be uniquely expressed product H. In this paper we obtain multiple results on codes finite groups, including fe...