Polytopes of high rank for the alternating groups

There is a well-known correspondence between abstract regular polytopes and string C-groups. In this paper, for each d?3, a string C-group with d generators, isomorphic to an alternating group of degree n is constructed (for some n?9), or equivalently an abstract regular d-polytope, is produced with automorphism group Alt(n). A method that extends the CPR graph of a polytope to a different CPR graph of a larger (or possibly isomorphic) polytope is used to prove that various groups are themselves string C-groups.     

6-Transitive graphs

A connected graph is n-transitive if, whenever two n-tuples are isometric, there is an automorphism mapping the first to the second. It is shown that a 6-transitive graph is complete multipartite, or complete bipartite with a matching deleted, or a cycle, or one of three special graphs on 9, 12 and 20 vertices. These graphs are n-transitive for all n; but there are graphs (the smallest on 56 vertices) which are 5- but not 6-transitive.     

Über symmetrische Graphen vom grad fünf

Let G be a connected 1-transitive graph of valency five. It is shown that the order of a vertex stabilizer divides 5 · 3² · 2¹⁷. A theorem of A. Gardiner bounding the order of a vertex stabilizer of a 2-transitive graph of valency 1 + p,p prime, is reproved.     

Automorphisms of graphs and coverings

We apply the theory of covering spaces to show how one can construct infinitely many finite s-transitive or locally s-transitive graphs. N. Biggs has used for similar purpose a special graph covering construction due to J. H. Conway.     

Finite polytopes have finite regular covers

We prove that any finite, abstract n-polytope is covered by a finite, abstract regular n-polytope.     

Pentavalent symmetric graphs of order 2p3

A graph is symmetric or 1-regular if its automorphism group is transitive or regular on the arc set of the graph, respectively. We classify the connected pentavalent symmetric graphs of order 2p~3 for each prime p. All those symmetric graphs appear as normal Cayley graphs on some groups of order 2p~3 and their automorphism groups are determined. For p = 3, no connected pentavalent symmetric graphs of order 2p~3 exist. However, for p = 2 or 5, such symmetric graph exists uniquely in each case. For p 7, the connected pentavalent symmetric graphs of order 2p~3 are all regular covers of the dipole Dip5 with covering transposition groups of order p~3, and they consist of seven infinite families; six of them are 1-regular and exist if and only if 5 |(p- 1), while the other one is 1-transitive but not 1-regular and exists if and only if 5 |(p ± 1). In the seven infinite families, each graph is unique for a given order.     

Tetravalent s-Transitive Graphs of Order 4p 2

Let s be a positive integer. A graph is s -transitive if its automorphism group is transitive on s-arcs but not on (s?+?1)-arcs. Let p be a prime. Zhou (Discrete Math 309:6081?C6086, 2009) classified tetravalent s-transitive graphs of order 4p. In this article a complete classification of tetravalent s-transitive graphs of order 4p ² is given.     

Complete subgraphs of the graphs of convex polytopes

It is shown that if three vertices of the graph G(l') of a convex 3-polytope P are chosen, then G(P) contains a refinement of the complete graph C₄ on four vertices, for which the three chosen vertices are principal (that is, correspond to vertices of C₄ in the refinement). In general, all four vertices may not be preassigned as principal. For dimensions d?4, simple (simplicial) d-polytopes are constructed whose graphs contain sets of three (four) vertices, which cannot all be principal in any refinement of C₄₊₁.     

Polyhedral Embeddings of Cayley Graphs

When is a Cayley graph the graph of a convex d-polytope? We show that while this is not always the case, some interesting finite groups have this property.     

Hexavalent (G,s)-transitive graphs

Let X be a finite simple undirected graph with a subgroup G of the full automorphism group Aut(X). Then X is said to be (G, s)-transitive for a positive integer s, if G is transitive on s-arcs but not on (s + 1)-arcs, and s-transitive if it is (Aut(X), s)-transitive. Let G _v be a stabilizer of a vertex v ∈ V (X) in G. Up to now, the structures of vertex stabilizers G _v of cubic, tetravalent or pentavalent (G, s)-transitive graphs are known. Thus, in this paper, we give the structure of the vertex stabilizers G _v of connected hexavalent (G, s)-transitive graphs.     

GQ(4, 2)-extensions, strongly regular case

A strongly regular locallyGQ(4, 2)-graph is a graph with parameters either (126, 45, 12, 8) or (190, 45, 12, 10). The existence and the uniqueness of the corresponding locallyGQ(4, 2)-graph in the first case are well known. We prove that theGQ(4, 2)-hyperoval on ten vertices either is the Petersen graph, or is the Möbius 5-prism, or consists of two (2, 3)-subgraphs connected by three edges. We obtain homogeneousGQ(4, 2)-solutions with a strongly regular point graph; in particular, this implies the negative answer to the question of F. Buekenhout concerning the existence of a locallyGQ(4, 2)-graph with the parameters (190, 45, 12, 10).     

A Note on Lights-Out-Puzzle: Parity-State Graphs

A state of a graph G is an assignment of 0 or 1 to each vertex of G. A move of a state consists of choosing a vertex and then switching the value of the vertex as well as those of its neighbors. Two states are said to be equivalent if one state can be changed to the other by a series of moves. A parity-state graph is defined to be a graph in which two states are equivalent if and only if the numbers of 1’s in the two states have the same parity. We characterize parity-state graphs and present some constructions of parity-state graphs together with applications. Among other things, it is proved that the one-skeleton of the 3-polytope obtained from a simple 3-polytope by cutting off all vertices is a parity-state graph.     

A ½-transitive graph of valency 4 with a nonsolvable group of automorphisms

A graph X is said to be ½-transitive if its automorphism group acts transively on the sets of its vertices and edges but intransitively on the set of its arcs. A construction of a ½-transitive graph of valency 4 and girth 6 with a nonsolvable group of automorphism is given. © 1997 John Wiley & Sons, Inc. J Graph Theory 25, 1: 133–138, 1997     

Extensions of regular polytopes with preassigned Schläfli symbol

We say that a (d+1)-polytope P is an extension of a polytope K if the facets or the vertex figures of P are isomorphic to K. The Schläfli symbol of any regular extension of a regular polytope is determined except for its first or last entry. For any regular polytope K we construct regular extensions with any even number as first entry of the Schläfli symbol. These extensions are lattices if K is a lattice. Moreover, using the so-called CPR graphs we provide a more general way of constructing extensions of polytopes.     

Disjoint edge paths between given vertices of a convex polytope

A Gale diagram technique is used to show that if d is positive integer greater than one, then there is a d-polytope P such that there are [$\text{2(d + 4)}\text{5}$] pairs of distinct vertices of P which cannot all be joined by disjoint paths in the graph of P.     

The height of faces of 3-polytopes

The height of a face in a 3-polytope is the maximum degree of the incident vertices of the face, and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices, or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then h can be arbitrarily large; so we assume the absence of pyramidal faces. In 1940, Lebesgue proved that every quadrangulated 3-polytope has h ≤ 11. In 1995, this bound was lowered by Avgustinovich and Borodin to 10. Recently, we improved it to the sharp bound 8. For plane triangulation without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that h ≤ 20 which bound is sharp. Later, Borodin (1998) proved that h ≤ 20 for all triangulated 3-polytopes. Recently, we obtained the sharp bound 10 for triangle-free 3-polytopes. In 1996, Horňák and Jendrol’ proved for arbitrarily 3-polytopes that h ≤ 23. In this paper we improve this bound to the sharp bound 20.     

Bartholdi zeta and L-functions of weighted digraphs, their coverings and products

Since a zeta function of a regular graph was introduced by Ihara [Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan 19 (1966) 219-235], many kinds of zeta functions and L-functions of a graph or a digraph have been defined and investigated. Most of the works concerning zeta and L-functions of a graph contain the following: (1) defining a zeta function, (2) defining an L-function associated with a (regular) graph covering, (3) providing their determinant expressions, and (4) computing the zeta function of a graph covering and obtaining its decomposition formula as a product of L-functions. As a continuation of those works, we introduce a zeta function of a weighted digraph and an L-function associated with a weighted digraph bundle. A graph bundle is a notion containing a cartesian product of graphs and a (regular or irregular) graph covering. Also we provide determinant expressions of the zeta function and the L-function. Moreover, we compute the zeta function of a weighted digraph bundle and obtain its decomposition formula as a product of the L-functions.     

Approximation Schemes for the Generalized Traveling Salesman Problem

A graph Γ is called a Deza graph if it is regular and the number of common neighbors of any two distinct vertices is one of two fixed values. A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular. In 1992, Gardiner et al. proved that a strongly regular graph that contains a vertex with disconnected second neighborhood is a complete multipartite graph with parts of the same size greater than 2. In this paper, we study strictly Deza graphs with disconnected second neighborhoods of vertices. In Section 2, we prove that, if each vertex of a strictly Deza graph has disconnected second neighborhood, then the graph is either edge-regular or coedge-regular. In Sections 3 and 4, we consider strictly Deza graphs that contain at least one vertex with disconnected second neighborhood. In Section 3, we show that, if such a graph is edge-regular, then it is the s-coclique extension of a strongly regular graph with parameters (n, k, λ, μ), where s is an integer, s ≥ 2, and λ = μ. In Section 4, we show that, if such a graph is coedge-regular, then it is the 2-clique extension of a complete multipartite graph with parts of the same size greater than or equal to 3.     

On a class of graphs without 3-stars

M. Numata described edge regular graphs without 3-stars. Allμ-subgraphs of these graphs are regular of the same valency. We prove that a connected graph without 3-stars all of whoseμ- subgraphs are regular of valencyα > 0 is either a triangular graph, or the Shläfli graph, or the icosahedron graph.     

Zeta functions of line, middle, total graphs of a graph and their coverings

We consider the (Ihara) zeta functions of line graphs, middle graphs and total graphs of a regular graph and their (regular or irregular) covering graphs. Let L(G), M(G) and T(G) denote the line, middle and total graph of G, respectively. We show that the line, middle and total graph of a (regular and irregular, respectively) covering of a graph G is a (regular and irregular, respectively) covering of L(G), M(G) and T(G), respectively. For a regular graph G, we express the zeta functions of the line, middle and total graph of any (regular or irregular) covering of G in terms of the characteristic polynomial of the covering. Also, the complexities of the line, middle and total graph of any (regular or irregular) covering of G are computed. Furthermore, we discuss the L-functions of the line, middle and total graph of a regular graph G.