Since is normal edge transitive, by theorem 1, acts transitively on or where is an orbit of the action 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. Introduction all graphs in this paper are assumed to be. That nis connected and vertex and edgetransitive follows from lemma2. Edgetransitive products virginia commonwealth university vcu.
Mingyao xu peking university vertextransitive and cayley graphs january 18, 2011 5 16. A graph is called vertextransitive or edgetransitive if the automorphism group aut, acts transitively on vertexset or edgeset of, respectively. 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. 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. This result generalizes the classification of doubly transitive balanced incomplete block designs with. Mar 28, 2014 the polycirculant conjecture asserts that every vertex transitive digraph has a semiregular automorphism. Regular coverings, edge transitive graphs, semisymmetric graphs, symmetric graphs. Arctransitive graph, a graph whose automorphism group acts transitively upon ordered pairs of adjacent vertices. Interestingly, we did not need to know much about the automorphism group.
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. Most primitive groups are full automorphism groups of edgetransitive hypergraphs. An edge automorphism is an edge isomorphism from a graph to itself. Automorphism groups of edgetransitive maps scdo 2016. Edgetransitive edgetransitive maps have12 or4orbits under the automorphism group edge type p,q. On cubic graphs admitting an edgetransitive solvable group.
Digraphs admitting sharply edgetransitive automorphism. In this paper, we classify all connected cubic edgetransitive graphs of order 4p3 for each prime p. Digraphs admitting sharply edgetransitive automorphism groups. 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. If the group acts transitively on edges, the graph is edgetransitive. See 1 for further properties and examples of halftransitive graphs. No block can be fixed by any nonidentity element of. If a graph is edgetransitive but not 1transitive thenanyedgecanbemappedtoany other, butinonlyoneofthetwopossible ways. We show that their reducible graphs in this family have quasi primitive automorphism groups, and prove using the classi. 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. 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. If a graph is edge transitive but not 1 transitive thenanyedgecanbemappedtoany other, butinonlyoneofthetwopossible ways. Digraphs admitting sharply edge transitive automorphism groups peter j.
If is connected normal edge transitive cayley graph, then is even, greater than 2, and is contained in. The basic graphs either admit an edgetransitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an. Also, cl3 is vertextransitive but not edge transitive, since six edges lie on a 3 cycle and three do not. Graphs with given automorphism group and few edge orbits. I give new constructions and nonexistence theorems for these. September 22, 1982 in this paper, we prove that every vertex transitive graph can be expressed as the edge disjoint union of symmetric graphs.
Digraphs admitting sharply edgetransitive automorphism groups peter j. Ebe an sedge transitive hypergraph such that dv 2 for every v 2v and jeij 2 for every ei 2e. Automorphism groups, isomorphism, reconstruction chapter 27. Let now h be a 2 edge transitive hypergraph, dregular and runiform, with d. The set of edgeautomorphisms forms a subgroup of the symmetric group on ex. 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. 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. If a graph g is edgetransitive, then it need not be vertextransitive.
Cayley graphs of abelian groups which are not normal edge. That nis connected and vertex and edge transitive follows from lemma2. Regular coverings, edgetransitive graphs, semisymmetric graphs, symmetric graphs. If y is a random subset of xand yg the set of gtranslates of y then. Semiregular automorphisms of edgetransitive graphs.
Intransitivity, properties of binary relations in mathematics. Leighton massachusetts institute of technology, cambridge, ma 029 accepted. Pdf finite edgetransitive oriented graphs of valency four. In my harvard senior thesis 2, i described a graph that is vertex transitive and edge transitive but not 1 transitive. On the automorphism groups of vertextransitive cayley. 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. The automorphism group of is the set of permutations of the vertex set that preserve adjacency. Let now h be a 2edge transitive hypergraph, dregular and runiform, with d. Edgetransitive tessellations with nonnegative euler. Semiregular automorphisms of edgetransitive graphs, journal. Examples 1 and 3 are transitive group actions, but example 2 is not. In this paper, we investigate the existence of semiregular automorphisms of edgetransitive graphs. Class 05c76, 05c25 1 introduction a graph gis vertextransitive resp. As a corollary, we determine which rose window graphs are vertextransitive.
A graph is vertextransitive if its automorphism group acts transitively on the set of vertices. A vertextransitive graph need not be edgetransitive. The polycirculant conjecture asserts that every vertextransitive digraph has a semiregular automorphism. In 2011 orbani c, pellicer, pisanski and tucker classi ed the edge transitive maps of low genus, together with those on e2.
This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. Research article normal edgetransitive cayley graphs of the. Most primitive groups are full automorphism groups of edge. An edgetransitive graph need not be vertex transitive, see p3. Edgetransitive products virginia commonwealth university. Basic fact every automorphism of a graph x induces a unique edgeautomorphism. Cameron digraphs having the property of the title were considered by babai, cameron, deza and sighi in 1981. Since is a subgroup of, thus or, in the latter case, should be contained in one of orbits of on.
In particular, we show that any regular edgetransitive graph of valency three or four has a semiregular automorphism. In this paper, we investigate the existence of semiregular automorphisms of edge transitive graphs. Complete multipartite graphs are especially interesting. In particular, we show that any regular edge transitive graph of valency three or four has a semiregular automorphism. Also using this characterization, we explicitly determine the automorphism groups of vertextransitive cayley digraphs of monoids. An edge transitive graph need not be vertex transitive, see p3. 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. Automorphism groups, isomorphism, reconstruction chapter. Let jxj n, and let gbe a primitive permutation group on xbut not the alternating group. Mingyao xu peking university vertex transitive and cayley graphs january 18, 2011 5 16. Graphs with given automorphism group and few edge orbits laszlo babal, albert j. Pdf a graph is said to be edgetransitive if its automorphism group acts transitively on its edges. September 22, 1982 in this paper, we prove that every vertextransitive graph can be expressed as the edgedisjoint union of symmetric graphs. Basic fact every automorphism of a graph x induces a unique edge automorphism.
A graph is vertex transitive if its automorphism group acts transitively on the set of vertices. On the decomposition of vertextransitive graphs into multicycles f. An edgeautomorphism is an edgeisomorphism from a graph to itself. On the decomposition of vertextransitive graphs into. Acayleygraphofagroup is called normal edge transitive 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. 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. The polycirculant conjecture asserts that every vertex transitive digraph has a semiregular automorphism. The group of order 7 is an automorphism group a subgroup of the full automorphism group. Above proposition cayley graphs are just those vertextransitive graphs whose full automorphism groups have a regular subgroup. This paper is a contribution to the study of the automorphism groups of designs. W n is neither vertextransitive nor edgetransitive for n 5 e the petersen graph is vertextransitive. If is complete, then any edge transitive group gis 2homogeneous on the vertices of that is, transitive on the unordered 2sets. All the points of the design lie in one orbit of, so the automorphism group is transitive on the points. If is connected normal edgetransitive cayley graph, then is even, greater than 2, and is contained in.
On connected tetravalent normal edgetransitive cayley. 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. A graph is called edge transitive if its automorphism group acts transitively on its edge set. 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. Introduction it is well known that, although every abstract group is the full automorphism group of a graph, not every permutation group is. Let be design and aut a block transitive and a point primitive. On classifying finite edge colored graphs with two transitive. For each of the 14 classes of edgetransitive maps described by graver and watkins, necessary and sufficient conditions are given for. Since is normal edgetransitive, by theorem 1, acts transitively on or where is an orbit of the action of. All transitive 2homogeneous groups are primitive see 5, page 35, so all of their nontrivial normal subgroups are transitive. Arc transitive graph, a graph whose automorphism group acts transitively upon ordered pairs of adjacent vertices. A 27vertex graph that is vertextransitive and edge. 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.
The set of edge automorphisms forms a subgroup of the symmetric group on ex. In this paper, we classify all connected cubic edge transitive graphs of order 4p3 for each prime p. Automorphism groups of edgetransitive maps request pdf. Cayley graphs of abelian groups which are not normal edgetransitive mehdi alaeiyan, hamid tavallaee, and ali a. Syllogism, a related notion in propositional logic. Graphs that are vertex transitive but not edge transitive. 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. Why the petersen graph is edge transitive mathematics stack. The automorphism groups of nonedgetransitive rose window. Above proposition cayley graphs are just those vertex transitive graphs whose full automorphism groups have a regular subgroup. As a corollary, we determine which rose window graphs are vertex transitive. If is complete, then any edgetransitive group gis 2homogeneous on the vertices of that is, transitive on the unordered 2sets. Akhil mathewprimes conference, may 2016 the outer automorphism of s6.
An automorphism of a graph is a permutation on the vertex set which preserves the edges of the graph. Every such automorphism can be embedded as the restriction to an invariant spine of some orientationpreserving periodic homeomorphism of a punctured surface. In this paper, we examine the structure of vertex and edgetransitive strongly regular graphs,using normal quotient reduction. If the group acts transitively on edges, the graph is edge transitive. 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. Clearly, arc transitive graphs are both vertextransitive and edgetransitive. In my harvard senior thesis 2, i described a graph that is vertextransitive and edgetransitive but not 1transitive. On the decomposition of vertex transitive graphs into multicycles f.
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