Abstract
A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs. Let p be a prime. A graph is called a p -graph if it is a Cayley graph of order a power of p. In this paper, a characterization is given of tetravalent edge-transitive p-graphs with p an odd prime. This is then applied to construct infinitely many connected tetravalent half-arc-transitive non-normal p-graphs with p an odd prime, and to initiate an investigation of tetravalent half-arc-transitive non-metacirculant p-graphs with p an odd prime. As by-products, two problems reported in the literature are answered.
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Notes
In ([11], Conjecture 3.5), it was conjectured that every tetravalent half-arc-transitive graph of prime power order has vertex stabilizers of order 2. This conjecture was first disproved by an example of order \(2^8=256\) given in [6] (We thank a referee for pointing out this). Our construction gives counterexamples to the above-mentioned conjecture for all odd primes.
\(\Gamma _{3,1}^\mathrm{nonnormal}\) is actually the smallest tetravalent half-arc-transitive non-normal p-graph for odd prime p. The author gratefully thanks Marston Conder for his help in proving this fact. It will be a topic in our forthcoming paper.
References
Alspach, B., Parsons, T.D.: A construction for vertex-transitive graphs. Can. J. Math. 34, 307–318 (1982)
Antončič, I., Šparl, P.: Classification of quartic half-arc-transitive weak metacirculants of girth at most 4. Discrete Math. 339, 931–945 (2016)
Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge University Press, Cambridge (1993)
Blackburn, N.: On prime-power groups with two generators. Math. Proc. Camb. Philos. Soc. 54, 327–337 (1958)
Bouwer, I.Z.: Vertex and edge transitive, but not 1-transitive, graphs. Can. Math. Bull. 13, 231–237 (1970)
Conder, M.D.E., Potočnik, P., Šparl, P.: Some recent discoveries about half-arc-transitive graphs. ARS Math. Contemp. 8, 149–162 (2015)
Fang, X.G., Li, C.H., Xu, M.-Y.: On edge-transitive Cayley graphs of valency four. Eur. J. Comb. 25, 1107–1116 (2004)
Feng, Y.-Q.: On vertex-transitive graphs of odd prime-power order. Discrete Math. 248, 265–269 (2002)
Feng, Y.-Q., Lu, Z.P., Xu, M.-Y.: Automorphism groups of Cayley digraphs. In: Koolen, J., Kwak, J.H., Xu, M.Y. (eds.) Applications of Group Theory to Combinatorics, pp. 13–25. Taylor & Francis Group, London (2008)
Feng, Y.Q., Wang, K., Zhou, C.: Tetravalent half-transitive graphs of order \(4p\). Eur. J. Comb. 28, 726–733 (2007)
Feng, Y.-Q., Kwak, J.H., Xu, M.-Y., Zhou, J.-X.: Tetravalent half-arc-transitive graphs of order \(p^4\). Eur. J. Combin. 29, 555–567 (2008)
Godsil, C.D.: On the full automorphism group of a graph. Combinatorica 1, 243–256 (1981)
Holt, D.F.: A graph which is edge transitive but not arc transitive. J. Graph Theory 5, 201–204 (1981)
Kutnar, K., Marušič, D., Šparl, P., Wang, R.-J., Xu, M.-Y.: Classification of half-arc-transitive graphs of order \(4p\). Eur. J. Combin. 34, 1158–1176 (2013)
Li, C.H., Lu, Z.P., Zhang, H.: Tetravalent edge-transitive Cayley graphs with odd number of vertices. J. Comb. Theory B 96, 164–181 (2006)
Li, C.H., Sim, H.S.: On half-transitive metacirculant graphs of prime-power order. J. Comb. Theory B 81, 45–57 (2001)
Li, C.H., Song, S.J., Wang, D.J.: A characterization of metacirculants. J. Comb. Theory A 120, 39–48 (2013)
Malnič, A., Marušič, D.: Constructing 4-valent \(\frac{1}{2}\)-transitive graphs with a non-solvable automorphism group. J. Comb. Theory B 75, 46–55 (1999)
Marušič, D.: Recent developments in half-transitive graphs. Discrete Math. 182, 219–231 (1998)
Marušič, D.: Half-transitive group actions on finite graphs of valency 4. J. Comb. Theory B. 73, 41–76 (1998)
Marušič, D.: Quartic half-arc-transitive graphs with large vertex stabilizers. Discrete Math. 299, 180–193 (2005)
Marušič, D., Šparl, P.: On quartic half-arc-transitive metacirculants. J. Algebr. Comb. 28, 365–395 (2008)
Marušič, D., Waller, A.: Half-transitive graphs of valency 4 with prescribed attachment numbers. J. Graph Theory 34, 89–99 (2000)
Potočnik, P., Spiga, P., Verret, G.: A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two. ARS Math. Contemp. 8, 133–148 (2015)
Robinson, D.J.: A Course in the Theory of Groups, 2nd edn. Springer, New York (1996)
Šparl, P.: A classification of tightly attached half-arc-transitive graphs of valency 4. J. Comb. Theory B 98, 1076–1108 (2008)
Šparl, P.: On the classification of quartic half-arc-transitive metacirculants. Discrete Math. 309, 2271–2283 (2009)
Šparl, P.: Almost all quartic half-arc-transitive weak metacirculants of Class II are of Class IV. Discrete Math. 310, 1737–1742 (2010)
Suzuki, M.: Group Theory I. Springer, New York (1982)
Tutte, W.T.: Connectivity in Graphs. University of Toronto Press, Toronto (1966)
Xu, M.-Y.: Half-transitive graphs of prime-cube order. J. Algebr. Comb. 1, 275–282 (1992)
Xu, M.-Y.: Automorphism groups and isomorphisms of Cayley digraphs. Discrete Math. 182, 309–319 (1998)
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11271029), and the Fundamental Research Funds for the Central Universities (2015JBM110). The author gratefully acknowledges the University of Western Australia for hospitality during his visit in 2014. The author also would like to thank the anonymous referees for the valuable comments and suggestions.
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Zhou, JX. Tetravalent half-arc-transitive p-graphs. J Algebr Comb 44, 947–971 (2016). https://doi.org/10.1007/s10801-016-0696-4
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DOI: https://doi.org/10.1007/s10801-016-0696-4