Half-arc-transitive graphs. with small number of alternets

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1 Half-arc-transitive graphs with small number of alternets University of Primorska, Koper, Slovenia This is a joint work with Ademir Hujdurović and Dragan Marušič. Villanova, June 2014

2 Overview Half-arc-transitive graphs Alternets Some results about the graphs with small number of alternets (2,3,4 or 5)

3 Different types of transitivity A graph is vertex-transitive if its automorphism group acts transitively on vertices.

4 Different types of transitivity A graph is vertex-transitive if its automorphism group acts transitively on vertices. A graph is edge-transitive if its automorphism group acts transitively on edges.

5 Different types of transitivity A graph is vertex-transitive if its automorphism group acts transitively on vertices. A graph is edge-transitive if its automorphism group acts transitively on edges. A graph is arc-transitive (also called symmetric) if its automorphism group acts transitively on arcs.

6 Different types of transitivity A graph is vertex-transitive if its automorphism group acts transitively on vertices. A graph is edge-transitive if its automorphism group acts transitively on edges. A graph is arc-transitive (also called symmetric) if its automorphism group acts transitively on arcs. A graph X is G-half-arc-transitive if the group G Aut(X ) acts transitively on the vertex set and the edge set of X, and G does not act transitively on the set of arcs of X. When X is Aut(X ) half-arc-transitive, then we shortly say that X is half-arc-transitive. We also say that a pair (G, X ) is half-arc-transitive.

7 Overview of the study of half-arc-transitive graphs The first result linking vertex-transitivity and edge-transitivity to arc-transitivity. Tutte, 1966 A vertex-transitive and edge-transitive graph of odd valency is arc-transitive. Half-arc-transitive graphs are of even valency.

8 The first examples of HAT graphs The first examples of half-arc-transitive graphs were given by Bouwer (1970), he constructed a 2k-valent half-arc-transitive graph for every k 2. The smallest example constructed by Bouwer had 54 vertices and was quartic. Dolye (1976) and Holt (1981) subsequently discovered the graph on 27 vertices, now known as the Doyle-Holt graph. In 1991 Alspach, Nowitz and Marušič showed this to be the smallest such graph.

9 The Doyle - Holt graph The smallest half-arc-transitive graph.

10 Current research on HAT graphs Papers dealing with the construction problem of HAT graphs, the classification problem of HAT graphs, of particular order or valency (Alspach, Conder, D Azevedo, Feng, Kwak, Li, Nedela, Malnič, Marušič, Pisanski, Praeger, Sim, Šajna, Šparl, Waller, Wang, Wilson, Zhou, Xu). There are several approaches that are currently being taken, such as for example, investigation of (im)primitivity of half-arc-transitive group actions on graphs, and geometry related questions about half-arc-transitive graphs.

11 Alternets Let X be a G-half-arc-transitive graph. Then we have a natural orientation of X. Declare two directed edges to be "related" if they have the same initial vertex, or the same terminal vertex. Consider the equivalence relation generated by "related". We will use the term alternet for the sub-graph consisting of an equivalence class of directed edges. When the graph is of valency 4, alternets are in fact cycles, and are called alterneting cycles. Alternets are blocks of imprimitivity for the action of G on the edges of X.

12 Alternets Given an arc (u, v) of D G (X ), the vertices u and v are called its tail and head, respectively. Let A = {A i i {1, 2,..., m}} be the corresponding set of alternets in X. The length of an alternet is the number of vertices contained in it. If there are at least two alternets, then all of them have the same even length, half of which is called the G-radius of X. For each i, define the head set H i to be the set of all vertices at the heads of arcs in A i A and the tail set T i to be the set of all vertices at the tails of arcs in A i A. Note that for m 2 we have T i H i =. The head sets as well as the tail sets partition V (X ), in particular, these are two G-invariant partitions of V (X ). The intersection A = A i,j = H i T j, when non-empty, is called a G-attachment set.

13 Example Example Let X 4 be a graph with vertex set V (X 4 ) = {(i, j) i Z 4, j Z 4 \ {i}} and edge set E(X 4 ) = {{(i, j), (k, i)} (i, j) V (X 4 ), k Z 4, k i, k j}. Then (S 4, X 4 ) is a half-arc-transitive with 4 alternets.

14 X 4

15 X 4

16 Alternets The collection of all attachment sets forms a G-invariant partition (intersection of two blocks is a block), consequently all attachment sets have the same cardinality, called the G-attachment number of X. If A i,j is a singleton then X is said to be G-loosely attached. On the other hand, if A i,j = H i = T j then X is said to be G-tightly attached. (In all of the above concepts the symbol G is omitted when the group G is clear from the context.)

17 Graphs with 2 alternets It is easy to see that every half-arc-transitive pair (G, X ) with 2 alternets is tightly attached. Such graphs are also necessarily bipartite. Example The graph K p,p pk 2, where p 1 (mod 4) is a prime (p 5), admits a half-arc-transitive group action with 2 alternets.

18 Graphs with 2 alternets It is easy to see that every half-arc-transitive pair (G, X ) with 2 alternets is tightly attached. Such graphs are also necessarily bipartite. Example The graph K p,p pk 2, where p 1 (mod 4) is a prime (p 5), admits a half-arc-transitive group action with 2 alternets. We have K p,p pk 2 = Cay(Z2p, {±1, ±3,..., ±(p 2)}). Define the mappings ρ, α : Z 2p Z 2p with ρ(x) = x + 2 and α(x) = ax + 1 where a is a generator of Z 2p. Straightforward calculations shows that the group G = ρ, α acts half-arc-transitively on X.

19 Graphs with 3 alternets Theorem (Hujdurović, KK, Marušič, JCTA, 2014) Let X be a G-half-arc-transitive graph with three alternets. Then X is G-tightly attached. An example of a half-arc-transitive graph with three alternets is the Doyle-Holt graph.

20 Examples of non-tightly-attached graphs Example Let n be a natural number n 4. Let X n be a graph with vertex set and edge set V (X n ) = {(i, j) i Z n, j Z n \ {i}} E(X n ) = {{(i, j), (k, i)} (i, j) V (X n ), k Z n, k i, k j}. Then (S n, X n ) is a half-arc-transitive with n alternets, which is not tightly attached.

21 An example of a graph with 4 alternets X 4

22 Graphs with 4 alternets Theorem (Hujdurović, KK, Marušič, JCTA, 2014) Let X be a G-half-arc-transitive graph with four alternets which is not G-tightly attached. Then there exists a decomposition of V (X ) relative to which the quotient graph of X is isomorphic to X 4 = R6 (5, 4).

23 Graphs with 4 alternets Theorem (Hujdurović, KK, Marušič, JCTA, 2014) Every connected regular cyclic cover of the rose window graph R 6 (5, 4), along which the half-arc-transitive subgroup of its automorphism group giving rise to four alternets lifts, is arc-transitive.

24 Graphs with 4 alternets Theorem (Hujdurović, KK, Marušič, JCTA, 2014) Every connected regular cyclic cover of the rose window graph R 6 (5, 4), along which the half-arc-transitive subgroup of its automorphism group giving rise to four alternets lifts, is arc-transitive. There exist a cover of the multigraph, obtained from X 4 by replacing each edge with two parallel edges, which is half-arc-transitive non-tightly attached with 4 alternets. Question Does there exist a regular cover of R 6 (5, 4) which is (non-tightly attached) half-arc-transitive with four alternets?

25 Graphs with 5 alternets Theorem (Hujdurović, KK, Marušič, JCTA, 2014) Let X be a G-half-arc-transitive graph with five alternets which is not G-tightly attached. Then there exists a decomposition of V (X ) relative to which the quotient graph of X is isomorphic to X 5.

26 Constructing graphs admitting HAT group actions with a prescribed number of alternets When aiming for constructions of graphs admitting half-arc-transitive group actions with a prescribed number of alternets, direct products of graphs prove useful. The direct product X Y of graphs X and Y has vertex set V (X ) V (Y ) where two vertices (x 1, y 1 ) and (x 2, y 2 ) are adjacent if and only if {x 1, x 2 } E(X ) and {y 1, y 2 } E(Y ). Let G and H be groups, and X and Y be a G-arc-transitive graph and a H-half-arc-transitive graph, respectively. Then it is not difficult to see that X Y is G H-half-arc-transitive graph. Furthermore, the direct product X Y is connected provided X and Y are connected and at least one of them is not bipartite.

27 Theorem (Hujdurović, KK, Marušič, JCTA, 2014) Let G and H be groups, and let X be a connected G-arc-transitive graph, and let Y be a connected H-half-arc-transitive graph with n 2 alternets. Then (i) if X is not bipartite, then X Y is a connected G H-half-arc-transitive graph with n alternets; and (ii) if X is bipartite and Y is not bipartite, then X Y is a connected G H-half-arc-transitive graph with 2n alternets. Remark: Since there are infinitely many non-bipartite arc-transitive graphs, this theorem implies the existence of infinitely many half-arc-transitive graphs with n 2 alternets provided one such graph exist.

28 Thank you!!!