Arc-Transitive Graph

An arc-transitive graph, sometimes also called a flag-transitive graph, is a graph whose graph automorphism group acts transitively on its graph arcs (Godsil and Royle 2001, p. 59).

More generally, a graph  is called -arc-transitive (or simply "-transitive") with  if it has an s-route and if there is always a graph automorphism of  sending each s-route onto any other -s-route (Harary 1994, p. 173). In other words, a graph is -transitive if its automorphism group acts transitively on all the s-routes (Holton and Sheehan 1993, p. 203). Note that various authors prefer symbols other than , for example  (Harary 1994, p. 173) or .

Arc-transitivity is an even stronger property than edge-transitivity or vertex-transitivity, so arc-transitive graphs have a very high degree of symmetry.

A 0-transitive graph is vertex-transitive. A 1-transitive graph is simply called an "arc-transitive graph" or even a "transitive graph." More confusingly still, arc-transitive graphs (and therefore in fact -transitive graphs for ) are sometimes called symmetric graphs (Godsil and Royle 2001, p. 59). This terminology conflict is particularly confusing since, as first shown by Bouwer (1970), graphs exist that are symmetric (in the sense of both edge- and vertex-transitive) but not arc-transitive, the smallest known example being the Doyle graph.

Symmetric non-arc-transitive graphs were first considered by Tutte (1966), who showed that any such graph must be regular of even degree. The first examples were given by Bouwer (1970), who gave a constructive proof for a connected -regular symmetric arc-intransitive graphs for all integers . The smallest such Bouwer graph has 54 vertices and is quartic. Another example of a symmetric non-arc-transitive graph is the 6-regular nonplanar diameter-3 graph on 111 vertices discovered by G. Exoo (E. Weisstein, Jul. 16, 2018).

A connected graph  with no endpoints (i.e., with minimum vertex degree ) is said to be strictly -transitive (with ) if  is -transitive but not -transitive (Holton and Sheehan 1993, p. 206). Such graphs have also been called -regular (Tutte 1947, Coxeter 1950, Frucht 1952) and -unitransitive (Harary 1994, p. 174). A strictly -transitive graph  has exactly one automorphism  such that  for any two -routes  and  of  (Harary 1994, p. 174).

The cycle graph  (for ) is -transitive for all , as is  for any positive integer  (Holton and Sheehan 1993, p. 204).

The numbers of arc-transitive graphs on , 2, ... vertices are 0, 1, 1, 3, 2, 6, 2, 8, 5, ... (OEIS A180240), as summarized in the table below, where  denotes a path graph,  a cycle graph,  is a ladder rung graph,  a complete graph,  a complete bipartite graph,  a complete tripartite graph,  a hypercube graph,  a circulant graph, and  a graph union of  copies of .

2
3
4	, ,
5	,
6	, , , octahedral graph , , utility graph
7	,
8	, , , cubical graph , , , 16-cell graph ,
9	, , , generalized quadrangle ,

The numbers of connected arc-transitive graphs on , 2, ... vertices are 0, 1, 1, 2, 2, 4, 2, 5, 4, 8, ... (OEIS A286280).

A tree may be -transitive yet not -transitive. For example, the star graph  with  is edge-transitive and 2-transitive, but not 1-transitive. However, an -transitive graph that is not a tree is also -transitive for all  (Holton and Sheehan 1993, p. 204), and so is most clearly termed "strictly -transitive."

The path graph  is -transitive (Holton and Sheehan 1993, p. 203), and a cycle graph  () is -transitive (Holton and Sheehan 1993, pp. 204 and 209, Exercise 6).

If  is an -transitive graph, then  is also -transitive for any  (Holton and Sheehan 1993, p. 204). But if  is disconnected and not the union of  copies of a single type of graph, then it is not vertex-transitive and hence not arc-transitive. Disconnected graphs therefore either have the same -transitivity as their identical connected components, or are not arc-transitive (if their components are not identical). The -transitivity of disconnected graphs is therefore trivial.

In 1947, Tutte showed that for any strictly -transitive connected cubic graph,  (Holton and Sheehan 1993, p. 207; Harary 1994, p. 175; Godsil and Royle 2001, p. 63). Weiss (1974) subsequently established the very deep result that for any regular connected strictly -transitive graph of degree ,  or  (Holton and Sheehan 1993, p. 208; Godsil and Royle 2001, p. 63).

If  is a vertex-transitive cubic graph on  vertices and  is its automorphism group, then if 3 divides the order of the stabilizer  of a vertex , then  is arc-transitive (Godsil and Royle 2001, p. 75).

Because there are no -transitive cubic graphs for , there are also no strictly -transitive ones (Harary 1994, p. 175). The 3-cages are strictly -transitive for  (Harary 1994, p. 175), but there also exist strictly -transitive graphs for  which are not cage graphs (Harary 1994, p. 175). These include the strictly 1-transitive graph of girth 12 on 432 nodes discovered by Frucht (1952) constructed as the Cayley graph of the permutations (2, 1, 5, 8, 3, 6, 7, 4, 9), (3, 6, 1, 4, 9, 2, 7, 8, 5), and (4, 3, 2, 1, 5, 7, 6, 8, 9) and now more commonly known as the cubic symmetric graph ; the strictly 2-transitive cubical, dodecahedral graphs, Möbius-Kantor graph , and Nauru graph; and the strictly 3-transitive Desargues graph  (Coxeter 1950). Some strictly -transitive graphs are illustrated above and summarized in the table below (partially based on the tables given by Coxeter 1950 and Harary 1994, p. 175).

			graph
1	432	3	cubic symmetric graph
2	4	3	tetrahedral graph
2	8	3	cubical graph
2	16	3	Möbius-Kantor graph
2	16	4	tesseract graph
2	20	3	dodecahedral graph
2	24	3	Nauru graph
2	32	5	5-hypercube graph
2	32	6	Kummer graph
2	64	6	6-hypercube graph
2	128	7	7-hypercube graph
2	256	8	8-hypercube graph
3	6	3	utility graph
3	20	3	Desargues graph
4	14	3	Heawood graph
5	30	3	Levi graph

REFERENCES

Bouwer, Z. "Vertex and Edge Transitive, But Not 1-Transitive Graphs." Canad. Math. Bull. 13, 231-237, 1970.

Conder, M. and Nedela, R. "Symmetric Cubic Graphs of Small Girth." J. Combin. Th. Ser. B 97, 757-768, 2007.

Conder, M. "All Symmetric Graphs of Order 2 to 30." Apr. 2014. https://www.math.auckland.ac.nz/~conder/symmetricgraphs-orderupto30.txt.

Coxeter, H. S. M. "Self-Dual Configurations and Regular Graphs." Bull. Amer. Math. Soc. 56, 413-455, 1950.

Doyle, P. G. On Transitive Graphs. Senior Thesis. Cambridge, MA, Harvard College, April 1976.

Doyle, P. "A 27-Vertex Graph That Is Vertex-Transitive and Edge-Transitive But Not L-Transitive." October 1998. http://hilbert.dartmouth.edu/~doyle/docs/bouwer/bouwer/bouwer.html.Frucht, R. "A One-Regular Graph of Degree Three." Canad. J. Math. 4, 240-247, 1952.

Gardiner, A. "Arc Transitivity in Graphs." Quart. J. Math. 24, 399-407, 1973.

Gardiner, A. "Arc Transitivity in Graphs II." Quart. J. Math. 25, 163-167, 1974.

Gardiner, A. "Arc Transitivity in Graphs III." Quart. J. Math. 27, 313-323, 1976.Godsil, C. and Royle, G. "Arc-Transitive Graphs." Ch. 4 in Algebraic Graph Theory. New York: Springer-Verlag, pp. 59-76, 2001.

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 174-175 and 200, 1994.

Holt, D. F. "A Graph Which Is Edge Transitive But Not Arc Transitive." J. Graph Th. 5, 201-204, 1981.

Holton, D. A. and Sheehan, J. The Petersen Graph. Cambridge, England: Cambridge University Press, pp. 202-210, 1993.

Lauri, J. and Scapellato, R. Topics in Graph Automorphisms and Reconstruction. Cambridge, England: Cambridge University Press, 2003.

Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 162 and 174, 1990.

Sloane, N. J. A. Sequences A180240 and A286280 in "The On-Line Encyclopedia of Integer Sequences."Tutte, W. T. "A Family of Cubical Graphs." Proc. Cambridge Philos. Soc. 43, 459-474, 1947.T

utte, W. T. Connectivity in Graphs. Toronto, CA: University of Toronto Press, 1966.Weiss, R. M. "Über -reguläre Graphen." J. Combin. Th. Ser. B 16, 229-233, 1974.