Groups of automorphisms of some graphs ijoar journals. Every such automorphism can be embedded as the restriction to an invariant spine of some orientationpreserving periodic homeomorphism of a punctured surface. Automorphism groups of edgetransitive maps request pdf. Digraphs admitting sharply edgetransitive automorphism groups. Semiregular automorphisms of edgetransitive graphs, journal. Also using this characterization, we explicitly determine the automorphism groups of vertextransitive cayley digraphs of monoids. Regular coverings, edge transitive graphs, semisymmetric graphs, symmetric graphs. An edge automorphism is an edge isomorphism from a graph to itself. Moreover, the alternating group is not the automorphism group of any family of sets, or. Edgetransitive products virginia commonwealth university vcu. The automorphism group of is the set of permutations of the vertex set that preserve adjacency. If a graph is edgetransitive but not 1transitive thenanyedgecanbemappedtoany other, butinonlyoneofthetwopossible ways. A 27vertex graph that is vertextransitive and edge. Graphs with given automorphism group and few edge orbits.
Automorphism groups, isomorphism, reconstruction chapter 27. An edge transitive graph need not be vertex transitive, see p3. The designs with unsolvable block transitive automorphism. If the group acts transitively on edges, the graph is edgetransitive.
If is connected normal edge transitive cayley graph, then is even, greater than 2, and is contained in. 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. Mingyao xu peking university vertex transitive and cayley graphs january 18, 2011 5 16. 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. 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. That nis connected and vertex and edge transitive follows from lemma2. Regular coverings, edgetransitive graphs, semisymmetric graphs, symmetric graphs. Pdf finite edgetransitive oriented graphs of valency four.
In 2011 orbani c, pellicer, pisanski and tucker classi ed the edgetransitive maps of low genus, together with those on e2. Let be design and aut a block transitive and a point primitive. Akhil mathewprimes conference, may 2016 the outer automorphism of s6. I give new constructions and nonexistence theorems for these. On classifying finite edge colored graphs with two transitive. Leighton massachusetts institute of technology, cambridge, ma 029 accepted. The polycirculant conjecture asserts that every vertextransitive digraph has a semiregular automorphism. 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. This paper is a contribution to the study of the automorphism groups of designs.
An automorphism of a graph is a permutation on the vertex set which preserves the edges of the graph. In this paper, we examine the structure of vertex and edgetransitive strongly regular graphs,using normal quotient reduction. 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. Edgetransitive edgetransitive maps have12 or4orbits under the automorphism group edge type p,q. If the group acts transitively on edges, the graph is edge transitive. Ebe an sedge transitive hypergraph such that dv 2 for every v 2v and jeij 2 for every ei 2e.
Why the petersen graph is edge transitive mathematics stack. Digraphs admitting sharply edge transitive automorphism groups peter j. Introduction all graphs in this paper are assumed to be. On classifying edge colored graphs with two transitive. A vertex transitive graph need not be edge transitive. All transitive 2homogeneous groups are primitive see 5, page 35, so all of their nontrivial normal subgroups are transitive. Interestingly, we did not need to know much about the automorphism group. Digraphs admitting sharply edgetransitive automorphism.
The set of edge automorphisms forms a subgroup of the symmetric group on ex. Graphs that are vertex transitive but not edge transitive. 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. In this paper, we classify all connected cubic edge transitive graphs of order 4p3 for each prime p. 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. Syllogism, a related notion in propositional logic.
On cubic graphs admitting an edgetransitive solvable group. Basic fact every automorphism of a graph x induces a unique edgeautomorphism. In my harvard senior thesis 2, i described a graph that is vertextransitive and edgetransitive but not 1transitive. 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 is complete, then any edgetransitive group gis 2homogeneous on the vertices of that is, transitive on the unordered 2sets. Introduction it is well known that, although every abstract group is the full automorphism group of a graph, not every permutation group is. Mingyao xu peking university vertextransitive and cayley graphs january 18, 2011 5 16. Since is normal edge transitive, by theorem 1, acts transitively on or where is an orbit of the action of. If a graph is edge transitive but not 1 transitive thenanyedgecanbemappedtoany other, butinonlyoneofthetwopossible ways. 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.
For each of the 14 classes of edgetransitive maps described by graver and watkins, necessary and sufficient conditions are given for. A graph is called edge transitive if its automorphism group acts transitively on its edge set. Semiregular automorphisms of edgetransitive graphs. Above proposition cayley graphs are just those vertex transitive graphs whose full automorphism groups have a regular subgroup. Pdf a graph is said to be edgetransitive if its automorphism group acts transitively on its edges. A vertextransitive graph need not be edgetransitive. 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. Talebi department of mathematics iran university of science and technology narmak, tehran 16844, iran received april 15, 2004 revised july 4, 2005 abstract.
We claim in the following proposition that every pair of edges intersects in at most one vertex. Ebe an s edge transitive hypergraph such that dv 2 for every v 2v and jeij 2 for every ei 2e. The set of edgeautomorphisms forms a subgroup of the symmetric group on ex. It also provides a classification of all doubly transitive symmetric association schemes. In this paper, we investigate the existence of semiregular automorphisms of edgetransitive graphs. Also, cl3 is vertextransitive but not edge transitive, since six edges lie on a 3 cycle and three do not. On the decomposition of vertex transitive graphs into multicycles f.
On connected tetravalent normal edgetransitive cayley. Clearly, arc transitive graphs are both vertextransitive and edgetransitive. 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. 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. Edgetransitive tessellations with nonnegative euler. Edgetransitive products virginia commonwealth university.
Automorphism groups, isomorphism, reconstruction chapter. 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. September 22, 1982 in this paper, we prove that every vertextransitive graph can be expressed as the edgedisjoint union of symmetric graphs. A graph is called edgetransitive if its automorphism group acts transitively on its edge set. Let jxj n, and let gbe a primitive permutation group on xbut not the alternating group. On the automorphism groups of vertextransitive cayley. See 1 for further properties and examples of halftransitive graphs. This result generalizes the classification of doubly transitive balanced incomplete block designs with. 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. Most primitive groups are full automorphism groups of edgetransitive hypergraphs. Arctransitive graph, a graph whose automorphism group acts transitively upon ordered pairs of adjacent vertices. Most primitive groups are full automorphism groups of edge.
Let now h be a 2edge transitive hypergraph, dregular and runiform, with d. Automorphism groups of edgetransitive maps scdo 2016. Cayley graphs of abelian groups which are not normal edgetransitive mehdi alaeiyan, hamid tavallaee, and ali a. On the decomposition of vertextransitive graphs into multicycles f. Above proposition cayley graphs are just those vertextransitive graphs whose full automorphism groups have a regular subgroup. Research article normal edgetransitive cayley graphs of the.
This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. An edgeautomorphism is an edgeisomorphism from a graph to itself. 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. If is connected normal edgetransitive cayley graph, then is even, greater than 2, and is contained in. 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. Since is a subgroup of, thus or, in the latter case, should be contained in one of orbits of on. The group of order 7 is an automorphism group a subgroup of the full automorphism group. The polycirculant conjecture asserts that every vertex transitive digraph has a semiregular automorphism. The automorphism groups of nonedgetransitive rose window. If y is a random subset of xand yg the set of gtranslates of y then. A graph is vertex transitive if its automorphism group acts transitively on the set of vertices. In this paper, we investigate the existence of semiregular automorphisms of edge transitive graphs.
Intransitivity, properties of binary relations in mathematics. 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. Cameron digraphs having the property of the title were considered by babai, cameron, deza and sighi in 1981. Mar 28, 2014 the polycirculant conjecture asserts that every vertex transitive digraph has a semiregular automorphism. The basic graphs either admit an edgetransitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an. Class 05c76, 05c25 1 introduction a graph gis vertextransitive resp. A graph is vertextransitive if its automorphism group acts transitively on the set of vertices. If a graph g is edgetransitive, then it need not be vertextransitive. 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. Class 05c76, 05c25 1 introduction a graph gis vertex transitive resp. An edgetransitive 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. Complete multipartite graphs are especially interesting. We show that their reducible graphs in this family have quasi primitive automorphism groups, and prove using the classi. Basic fact every automorphism of a graph x induces a unique edge automorphism. A graph is called vertextransitive or edgetransitive if the automorphism group aut, acts transitively on vertexset or edgeset of, respectively. Let now h be a 2 edge transitive hypergraph, dregular and runiform, with d. Cayley graphs of abelian groups which are not normal edge. Digraphs admitting sharply edgetransitive automorphism groups peter j. In my harvard senior thesis 2, i described a graph that is vertex transitive and edge transitive but not 1 transitive. 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.
Arc transitive graph, a graph whose automorphism group acts transitively upon ordered pairs of adjacent vertices. That nis connected and vertex and edgetransitive follows from lemma2. 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. As a corollary, we determine which rose window graphs are vertextransitive. Since is normal edgetransitive, by theorem 1, acts transitively on or where is an orbit of the action of. In 2011 orbani c, pellicer, pisanski and tucker classi ed the edge transitive maps of low genus, together with those on e2. W n is neither vertextransitive nor edgetransitive for n 5 e the petersen graph is vertextransitive. As a corollary, we determine which rose window graphs are vertex transitive. In particular, we show that any regular edge transitive graph of valency three or four has a semiregular automorphism. 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.
1120 1000 257 1106 326 1013 1394 1532 529 347 831 840 1126 274 1493 184 494 70 600 535 923 1431 968 1077 138 1283 339 900 854 1020 1041 13 1217 946 9 238 865 269 913 307 1239 1113 1257 781