Throughout this paper G is a finite group and S is a subgroup of G. When S is contained in the subgroup H of G, we say S is strongly closed in H with respect to G if whenever s ∈ S and g ∈ G are such that s ∈ H, then s ∈ S. In other words, every G-conjugacy class of elements of S intersected with H is contained in S. In the case where S is a p-group for some prime p we say that S is strongly cl...

A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...

A minor-closed class of graphs is a set of labelled graphs which is closed under isomorphism and under taking minors. For a minor-closed class G, we let gn be the number of graphs in G which have n vertices. A recent result of Norine et al. [12] shows that for all minor-closed class G, there is a constant c such that gn ≤ c nn!. Our main results show that the growth rate of gn is far from arbit...

25 The Traveling Salesman Problem (TSP) is still one of the most researched topics in computational mathematics, and we introduce a variant of it, namely the study of the closed k-walks in graphs. We 2 G. Bullington, R. Gera, L. Eroh And S.J. Winters search for a shortest closed route visiting k cities in a non complete graph without weights. This motivates the following definition. Given a set...

If a closed smooth n-manifold M admits a finite cover M̂ whose Z/2Z-cohomology has the maximal cup-length, then for any riemannian metric g on M , we show that the systole Sys(M, g) and the volume Vol(M, g) of the riemannian manifold (M, g) are related by the following isosystolic inequality: Sys(M, g) n ≤ n!Vol(M, g). The inequality can be regarded as a generalization of Burago and Hebda’s ineq...

We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed, or algebraically closed field F , we give a sufficient condition for a valued subfield of the field of generalized power series F ((G)) to admit a K-valuation basis. We show that the field of rational functions F (G) and the field F (G) of power series in F ((G)) algebraic ov...

We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed or algebraically closed field F with subfield K, we give a sufficient condition for a valued subfield of the field of generalized power series F ((G)) to admit aK-valuation basis. We show that the field of rational functions F (G) and the field F (G) of power series in F ((G))...

Two vertices u and v in a nontrivial connected graph G are twins if u and v have the same neighbors in V (G) − {u, v}. If u and v are adjacent, they are referred to as true twins; while if u and v are nonadjacent, they are false twins. For a positive integer k, let c : V (G) → Zk be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex c...

Let G=(V, E) be an Eulerian graph embedded on a triangulizable surface S. We show that E can be decomposed into closed curves C1 , ..., Ck such that mincr(G, D)= i=1 mincr(Ci , D) for each closed curve D on S. Here mincr(G, D) denotes the minimum number of intersections of G and D$ (counting multiplicities), where D$ ranges over all closed curves D$ freely homotopic to D and not intersecting V....