Neighborly maps with few vertices

A neighborly map is a simplicial 2-complex which decomposes a closed 2-manifold without boundary, such that any two vertices are joined by an edge (1-cell) in the complex. We find and describe all the neighborly maps with Euler characteristicX>−10 (i.e., genusg<6, if orientable) or, equivalently, all the neighborly maps withV<12 vertices.     

Combinatorial 3-manifolds with few vertices

It is proved that every combinatorial 3-manifold with at most eight vertices is a combinatorial sphere.     

The unique 3-neighborly 4-manifold with few vertices

Triangulated 4-dimensional manifolds with n vertices are considered such that any triple of vertices determines a triangle belonging to the triangulation. It is shown that for n ? 13 there is exactly one such triangulation (besides the boundary of a 5-simplex). It has nine vertices and it is a triangulation of the complex projective plane.     

On the maximal volume of convex bodies with few vertices

The problem of determining the largest volume of a (d + 2)-point set in E^d of unit diameter is settled. The extremal polytopes are described completely.     

3-trees with few vertices of degree 3 in circuit graphs

A circuit graph(G,C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface F_χ with Euler characteristic χ≥0 has a 3-tree with at most vertices of degree 3, where c_χ is a constant depending only on F_χ.     

Finite groups whose conjugacy class graphs have few vertices

In this note we classify the finite groups satisfying the following property P₅: their conjugacy class lengths are set-wise relatively prime for any 5 distinct classes.Received: 6 October 2004; revised: 16 November 2004     

Polynomial recognition of equal unions in hypergraphs with few vertices of large degree

A family of nonempty sets has the equal union property if there exist two nonempty disjoint subfamilies having equal unions. If every point belongs to the unions, then we say the family has the full equal union property. Recognition of both properties is NP-complete even when restricted to families for which the degree of every point is at most three. In this paper we show that both recognition problems can be solved in polynomial time for families in which there is a bound on the number of points whose degree exceeds two.     

Spanning trees whose reducible stems have a few branch vertices

Czechoslovak Mathematical Journal - Let T be a tree. Then a vertex of T with degree one is a leaf of T and a vertex of degree at least three is a branch vertex of T. The set of leaves of T is...     

Facets with fewest vertices

Forv>d≧3, letm(v, d) be the smallest numberm, such that every convexd-polytope withv vertices has a facet with at mostm vertices. In this paper, bounds form(v, d) are found; in particular, for fixedd≧3, $$\frac{{r - 1}}{r} \leqslant \mathop {\lim \inf }\limits_{\upsilon \to \infty } \frac{{m(\upsilon ,d)}}{\upsilon } \leqslant \mathop {\lim \sup }\limits_{\upsilon \to \infty } \frac{{m(\upsilon ,d)}}{\upsilon } \leqslant \frac{{d - 3}}{{d - 2}}$$ , wherer=[1/3(d+1)].     

Manifolds with Cusps

In this paper manifolds with cusps are defined and their properties investigated. Manifolds with cusps are manifolds with boundary where the dimension of the tangent cone at points varies widely with that point. Examples are exponential sums, splines, and polynomials with only real roots. The investigations are motivated by numerical algorithms in approximation theory and other areas.     

Manifolds with involution

In this paper, we determine the groups (k _i are odd), (k _i are odd and (k _i are even andn>k _l ), (k _i are even andn>k _l ), (k _i are even andn>k _l ,k _l 12),J _n ^1,2,J _n ^2,3,J _n ^1,4. And we obtain the relation Im _n ^k =J _n ^l,k .     

Specht modules with abelian vertices

In this article, we consider indecomposable Specht modules with abelian vertices. We show that the corresponding partitions are necessarily p ²-cores where p is the characteristic of the underlying field. Furthermore, in the case of p≥3, or p=2 and μ is 2-regular, we show that the complexity of the Specht module S ^μ is precisely the p-weight of the partition μ. In the latter case, we classify Specht modules with abelian vertices. For some applications of the above results, we extend a result of M. Wildon and compute the vertices of the Specht module S^(pp)S^{(p^{p})} for p≥3.     

Graphs with given neighborhoods of vertices

This article establishes criteria for existence of finite graphs L in which the neighborhoods of all vertices are isomorphic to a given graph M for the cases in which M is a simple arc and a simple cycle.Translated from Matematicheskie Zametki, Vol. 3, No. 2, pp. 211–216, February, 1968.