Name
Abstract | ||
|---|---|---|
A subgroup G of automorphisms of a graph X is said to be ½-arc-transitive if it is vertex- and edge- but not arc-transitive. The graph X is said to be ½-arc-transitive if Aut X is ½-arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and ½-arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and ½-arc-transitive group actions on graphs of valency 4 given via the well known concept of medial graphs (European J. Combin. 19 (1998) 345) is generalised. Any orientably regular hypermap H gives rise to a uniquely determined medial map whose underlying graph Y admits a ½-arc-transitive group action of the automorphism group G of the original hypermap H. Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G ≤ Aut X acting ½-arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is ½-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated ½-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of ½-arc-transitive group action. Indeed, in last section we have applied general results to construct ½-arc-transitive graphs with cycle stabilisers of arbitrarily large orders. |
Year | DOI | Venue |
|---|---|---|
2004 | 10.1016/j.ejc.2003.07.003 | Eur. J. Comb. |
Keywords | Field | DocType |
corresponding hypermap,graph x,chiral hypermaps,medial graph,arc-transitive group action,arc-transitive graph,orientably regular hypermap h,underlying graph y,aut x,half-arc-transitive graph,automorphism group,large order,concept map,group action | Graph automorphism,Discrete mathematics,Strongly regular graph,Circulant graph,Combinatorics,Vertex-transitive graph,Line graph,Edge-transitive graph,Regular graph,Symmetric graph,Mathematics | Journal |
Volume | Issue | ISSN |
25 | 3 | 0195-6698 |
Citations | PageRank | References |
5 | 0.49 | 9 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Antonio Breda d'Azevedo | 1 | 19 | 5.47 |
| Roman Nedela | 2 | 392 | 47.78 |