## Constructing infinite one-regular graphs.(English)Zbl 0943.05044

A graph $$X$$ is said to be one-regular if the automorphism group of $$X$$ acts regularly on the set of arcs of $$X$$. The authors consider infinite one-regular graphs. Starting with an infinite family of finite one-regular graphs of valency 4, for each member of this family an infinite one-regular graph is constructed. These graphs are Cayley graphs of almost abelian groups and represent a subclass of graphs with polynomial growth.

### MSC:

 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)

### Keywords:

automorphism group; one-regular graphs; Cayley graphs
Full Text:

### References:

 [1] Alspach, B.; Marušič, D.; Nowitz, L., Constructing graphs which are$$12$$-transitive, J. Aust. Math. Soc. A, 56, 391-402, (1994) · Zbl 0808.05057 [2] Conder, M. D.E.; Praeger, C. E., Remarks on path-transitivity, Europ. J. Combinatorics, 17, 371-378, (1996) · Zbl 0871.05029 [3] Djoković, D.Ž., Automorphisms of graphs and coverings, J. Comb. Theory, Ser. B, 16, 243-247, (1974) [4] Frucht, R., A one-regular graph of degree three, Can. J. Math., 4, 240-247, (1952) · Zbl 0046.40903 [5] Furst, M. L.; Gross, J. L.; McGeoch, L., Finding a maximal genus graph embedding, J. Assoc. Comput. Mach., 35, 523-534, (1988) [6] Gardiner, A.; Praeger, C. E., On 4-valent symmetric graphs, Europ. J. Combinatorics, 15, 375-381, (1995) · Zbl 0806.05037 [7] Gardiner, A.; Praeger, C. E., A characterization of certain families of 4-valent symmetric graphs, Europ. J. Combinatorics, 15, 383-397, (1995) · Zbl 0806.05038 [8] Godsil, C.; Seifter, N., Graphs of polynomial growth are covering graphs, Graphs Comb., 8, 233-241, (1992) · Zbl 0764.05034 [9] Gross, J. L.; Tucker, T. W., Topological Graph Theory, (1987), Wiley-Interscience New York [10] Imrich, W.; Seifter, N., A survey on graphs with polynomial growth, Discrete Math., 95, 101-117, (1991) · Zbl 0761.05048 [11] Marušič, D., A family of one-regular graphs of valency 4, Europ. J. Combinatorics, 18, 59-64, (1997) · Zbl 0870.05030 [12] Miller, R. C., The trivalent symmetric graphs of girth at most six, J. Comb. Theory, Ser. B, 10, 163-182, (1971) · Zbl 0223.05113 [13] Passman, D. S., Permutation Groups, (1968), Benjamin Menlo Park, California · Zbl 0164.33805 [14] Seifter, N., Properties of graphs with polynomial growth, J. Comb. Theory, Ser. B, 52, 222-235, (1991) · Zbl 0668.05034 [15] Sims, C. C., Graphs and finite permutation groups, Math. Z, 95, 76-86, (1967) · Zbl 0244.20001 [16] Sims, C. C., Graphs and finite permutation groups II, Math. Z, 103, 276-281, (1968) · Zbl 0259.20003 [17] Trofimov, V. I., Automorphisms of graphs and a characterization of lattices, Math. USSR Izvestiya, 22, 379-391, (1984) · Zbl 0534.05034 [18] Trofimov, V. I., Graphs with polynomial growth, Math. USSR Sbornik, 51, 405-417, (1985) · Zbl 0565.05035 [19] Tutte, W. T., A family of cubical graphs, Proc. Camb. Phil. Soc., 43, 459-474, (1948) · Zbl 0029.42401 [20] Weiss, R., The nonexistence of 8-transitive graphs, Combinatorica, 1, 309-311, (1981) · Zbl 0486.05032 [21] Wielandt, H., Finite Permutation Groups, (1964), Academic Press New York · Zbl 0138.02501 [22] Xuong, N. H., How to determine the maximum genus of a graph, J. Comb. Theory, Ser. B, 26, 217-225, (1979) · Zbl 0403.05035
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