Also using this characterization, we explicitly determine the automorphism groups of vertextransitive cayley digraphs of monoids. Semiregular automorphisms of edgetransitive graphs, journal. Akhil mathewprimes conference, may 2016 the outer automorphism of s6. Automorphism groups, isomorphism, reconstruction chapter. For each of the 14 classes of edgetransitive maps described by graver and watkins, necessary and sufficient conditions are given for a group to be the automorphism group of a map, or of an orientable map without boundary, in that class. Syllogism, a related notion in propositional logic. In this paper, we classify all connected cubic edgetransitive graphs of order 4p3 for each prime p. Graphs that are vertex transitive but not edge transitive. Edgetransitive tessellations with nonnegative euler. Research article normal edgetransitive cayley graphs of the. Pdf finite edgetransitive oriented graphs of valency four. For each of the 14 classes of edge transitive maps described by graver and watkins, necessary and sufficient conditions are given for a group to be the automorphism group of a map, or of an orientable map without boundary, in that class. Basic fact every automorphism of a graph x induces a unique edge automorphism. Ebe an sedge transitive hypergraph such that dv 2 for every v 2v and jeij 2 for every ei 2e.
In particular, we show that any regular edge transitive graph of valency three or four has a semiregular automorphism. On the decomposition of vertex transitive graphs into multicycles f. The polycirculant conjecture asserts that every vertextransitive digraph has a semiregular automorphism. A graph is called edge transitive if its automorphism group acts transitively on its edge set. Digraphs admitting sharply edgetransitive automorphism groups peter j. In this paper, we classify all connected cubic edge transitive graphs of order 4p3 for each prime p. An automorphism of a graph is a permutation on the vertex set which preserves the edges of the graph. On the decomposition of vertextransitive graphs into multicycles f. In this paper, we examine the structure of vertex and edgetransitive strongly regular graphs,using normal quotient reduction.
If a graph is edgetransitive but not 1transitive thenanyedgecanbemappedtoany other, butinonlyoneofthetwopossible ways. Talebi department of mathematics iran university of science and technology narmak, tehran 16844, iran received april 15, 2004 revised july 4, 2005 abstract. Most primitive groups are full automorphism groups of edge. Digraphs admitting sharply edge transitive automorphism groups peter j. Cayley graphs of abelian groups which are not normal edge. On classifying edge colored graphs with two transitive. Why the petersen graph is edge transitive mathematics stack. Mar 28, 2014 the polycirculant conjecture asserts that every vertex transitive digraph has a semiregular automorphism. Graphs with given automorphism group and few edge orbits. Class 05c76, 05c25 1 introduction a graph gis vertex transitive resp. An edge transitive graph need not be vertex transitive, see p3. In particular, we show that any regular edgetransitive graph of valency three or four has a semiregular automorphism. Automorphism groups of edgetransitive maps request pdf.
In my harvard senior thesis 2, i described a graph that is vertex transitive and edge transitive but not 1 transitive. If is connected normal edge transitive cayley graph, then is even, greater than 2, and is contained in. An edgeautomorphism is an edgeisomorphism from a graph to itself. Clearly, arc transitive graphs are both vertextransitive and edgetransitive. A graph is vertextransitive if its automorphism group acts transitively on the set of vertices. I give new constructions and nonexistence theorems for these. In 2011 orbani c, pellicer, pisanski and tucker classi ed the edgetransitive maps of low genus, together with those on e2. Request pdf automorphism groups of edge transitive maps for each of the 14 classes of edge transitive maps described by graver and watkins, necessary and sufficient conditions are given for a. As the full automorphism groups of edgetransitive rose window graphs have been determined, this will complete the problem of calculating the full automorphism group of rose window graphs. A graph is called vertextransitive or edgetransitive if the automorphism group aut, acts transitively on vertexset or edgeset of, respectively. Automorphism groups of edgetransitive maps scdo 2016. Intransitivity, properties of binary relations in mathematics.
We show that their reducible graphs in this family have quasi primitive automorphism groups, and prove using the classi. Using this unique prime factorization theorem we prove that if a graph x can be writteni as a product of connected rooted graphs, which are pairwise relatively prime, then the automorphism. Basic fact every automorphism of a graph x induces a unique edgeautomorphism. That nis connected and vertex and edgetransitive follows from lemma2. If a graph g is edgetransitive, then it need not be vertextransitive. If the group acts transitively on edges, the graph is edgetransitive. Automorphism groups, isomorphism, reconstruction chapter 27. No block can be fixed by any nonidentity element of. Request pdf automorphism groups of edgetransitive maps for each of the 14 classes of edgetransitive maps described by graver and watkins, necessary and sufficient conditions are given for a. Let jxj n, and let gbe a primitive permutation group on xbut not the alternating group. Transitive relation, a binary relation in which if a is related to b and b is related to c, then a is related to c. As a corollary, we determine which rose window graphs are vertex transitive. The automorphism groups of nonedgetransitive rose window. All the points of the design lie in one orbit of, so the automorphism group is transitive on the points.
Cameron digraphs having the property of the title were considered by babai, cameron, deza and sighi in 1981. Mingyao xu peking university vertextransitive and cayley graphs january 18, 2011 5 16. Let be design and aut a block transitive and a point primitive. Since is normal edgetransitive, by theorem 1, acts transitively on or where is an orbit of the action of. This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. Edgetransitive products virginia commonwealth university vcu. This result generalizes the classification of doubly transitive balanced incomplete block designs with. Above proposition cayley graphs are just those vertex transitive graphs whose full automorphism groups have a regular subgroup. In this paper, we investigate the existence of semiregular automorphisms of edgetransitive graphs. Request pdf on classifying edge colored graphs with two transitive automorphism groups this paper classifies all finite edge colored graphs with doubly transitive automorphism groups. The basic graphs either admit an edgetransitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an. Arc transitive graph, a graph whose automorphism group acts transitively upon ordered pairs of adjacent vertices. Most primitive groups are full automorphism groups of edgetransitive hypergraphs.
Cayley graphs of abelian groups which are not normal edgetransitive mehdi alaeiyan, hamid tavallaee, and ali a. Digraphs admitting sharply edgetransitive automorphism. Ebe an s edge transitive hypergraph such that dv 2 for every v 2v and jeij 2 for every ei 2e. Pdf a graph is said to be edgetransitive if its automorphism group acts transitively on its edges. Let now h be a 2edge transitive hypergraph, dregular and runiform, with d. Edgetransitive edgetransitive maps have12 or4orbits under the automorphism group edge type p,q. Since is normal edge transitive, by theorem 1, acts transitively on or where is an orbit of the action of. In 1998, xu conjectured that almost all cayl finally, in 1994, praeger and mckay conjectured that almost all undirected vertextransitive graphs are cayley graphs of groups. If the group acts transitively on edges, the graph is edge transitive.
September 22, 1982 in this paper, we prove that every vertextransitive graph can be expressed as the edgedisjoint union of symmetric graphs. Interestingly, we did not need to know much about the automorphism group. If is complete, then any edge transitive group gis 2homogeneous on the vertices of that is, transitive on the unordered 2sets. If is complete, then any edgetransitive group gis 2homogeneous on the vertices of that is, transitive on the unordered 2sets. A graph is vertex transitive if its automorphism group acts transitively on the set of vertices. Every such automorphism can be embedded as the restriction to an invariant spine of some orientationpreserving periodic homeomorphism of a punctured surface.
An automorphism of a graph is edge transitive if it acts transitively on the set of geometric edges components of the complement of the vertices, or equivalently, if there is no nontrivial invariant subgraph. In the mathematical field of graph theory, an edgetransitive graph is a graph g such that, given any two edges e 1 and e 2 of g, there is an automorphism of g that maps e 1 to e 2. Above proposition cayley graphs are just those vertextransitive graphs whose full automorphism groups have a regular subgroup. On classifying finite edge colored graphs with two transitive. Since is a subgroup of, thus or, in the latter case, should be contained in one of orbits of on. Acayleygraphofagroup is called normal edgetransitive if the normalizer of the right representation of the group in the automorphism of the cayley graph acts transitively on the set of edges of the graph. The set of edge automorphisms forms a subgroup of the symmetric group on ex. In this paper, we investigate the existence of semiregular automorphisms of edge transitive graphs. Regular coverings, edge transitive graphs, semisymmetric graphs, symmetric graphs.
On the automorphism groups of vertextransitive cayley. Regular coverings, edgetransitive graphs, semisymmetric graphs, symmetric graphs. A vertex transitive graph need not be edge transitive. On the decomposition of vertextransitive graphs into. W n is neither vertextransitive nor edgetransitive for n 5 e the petersen graph is vertextransitive. In the mathematical field of graph theory, an edge transitive graph is a graph g such that, given any two edges e 1 and e 2 of g, there is an automorphism of g that maps e 1 to e 2. On cubic graphs admitting an edgetransitive solvable group. A vertextransitive graph need not be edgetransitive. If y is a random subset of xand yg the set of gtranslates of y then. For such a graph, the automorphisms mapping an edge xyto uv are either always x u and y v, or always x v and y u.
In my harvard senior thesis 2, i described a graph that is vertextransitive and edgetransitive but not 1transitive. Complete multipartite graphs are especially interesting. As the full automorphism groups of edge transitive rose window graphs have been determined, this will complete the problem of calculating the full automorphism group of rose window graphs. In 2011 orbani c, pellicer, pisanski and tucker classi ed the edge transitive maps of low genus, together with those on e2. On connected tetravalent normal edgetransitive cayley. It also provides a classification of all doubly transitive symmetric association schemes. In a recent paper we showed that every connected graph can be written as a weak cartesian product of a family of indecomposable rooted graphs and that this decomposition is unique to within isomorphisms. Leighton massachusetts institute of technology, cambridge, ma 029 accepted. The automorphism group of is the set of permutations of the vertex set that preserve adjacency. You should be able to show something stronger as easily, that every automorphism which moves an edge in g corresponds to an automorphism of lg that moves a vertex, and vice versa. Let now h be a 2 edge transitive hypergraph, dregular and runiform, with d. Graphs with given automorphism group and few edge orbits laszlo babal, albert j.
Goodman and laszlo lovasz a graph x is said to represent the group g with k edge vertex orbits if the automorphism group of x is isomorphic to g and it acts with k orbits on the set of edges vertices, resp. Groups of automorphisms of some graphs ijoar journals. Arctransitive graph, a graph whose automorphism group acts transitively upon ordered pairs of adjacent vertices. Class 05c76, 05c25 1 introduction a graph gis vertextransitive resp. The group of order 7 is an automorphism group a subgroup of the full automorphism group. A graph is called edgetransitive if its automorphism group acts transitively on its edge set. See 1 for further properties and examples of halftransitive graphs. Moreover, the alternating group is not the automorphism group of any family of sets, or. The set of edgeautomorphisms forms a subgroup of the symmetric group on ex. Edgetransitive products virginia commonwealth university. If g is edge and vertex transitive it is possible that g is not arc transitive, that is, if ab is an edge, then there may be no automorphism. We claim in the following proposition that every pair of edges intersects in at most one vertex. Semiregular automorphisms of edgetransitive graphs. Introduction it is well known that, although every abstract group is the full automorphism group of a graph, not every permutation group is.
Also, cl3 is vertextransitive but not edge transitive, since six edges lie on a 3 cycle and three do not. A 27vertex graph that is vertextransitive and edge. The polycirculant conjecture asserts that every vertex transitive digraph has a semiregular automorphism. If a graph is edge transitive but not 1 transitive thenanyedgecanbemappedtoany other, butinonlyoneofthetwopossible ways. Examples 1 and 3 are transitive group actions, but example 2 is not. If is connected normal edgetransitive cayley graph, then is even, greater than 2, and is contained in. An edgetransitive graph need not be vertex transitive, see p3. That nis connected and vertex and edge transitive follows from lemma2. Introduction all graphs in this paper are assumed to be. An edge automorphism is an edge isomorphism from a graph to itself. Digraphs admitting sharply edgetransitive automorphism groups. September 22, 1982 in this paper, we prove that every vertex transitive graph can be expressed as the edge disjoint union of symmetric graphs.
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