Abstract
Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors (see [3], [8]–[15]). A covering of G is called circulant if its covering graph is circulant. Recently, the authors [4] enumerated the isomorphism classes of circulant double coverings of a certain kind, called typical, and showed that no double covering of a circulant graph of valency 3 is circulant. In this paper, the isomorphism classes of connected circulant double coverings of a circulant graph of valency 4 are enumerated. As a consequence, it is shown that no double covering of a non-circulant graph G of valency 4 can be circulant if G is vertex-transitive or G has a prime power of vertices.
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Bermond, J.-C., Comellas, F., Hsu, D.F.: Distributed loop computer networks: a survey. J. Parallel Distrib. Comput. 24, 2–10 (1995)
Du, S.F., Marušič, D., Waller, A.O.: On 2-arc-transitive covers of complete graphs. J Combin. Theory B 74, 376–390 (1998)
Feng, R., Kwak, J.H., Kim, J., Lee, J.: Isomorphism classes of concrete graph coverings. SIAM J. Disc. Math. 11, 265–272 (1998)
Feng, R., Kwak, J.H.: Typical circulant double coverings of a circulant graph. Discrete Math. 277, 73–85 (2004)
Godsil, C.D., Hensel, A.D.: Distance-regular covers of the complete graph. J Combin. Theory B 56, 205–238 (1992)
Gross, J.L., Tucker, T.W.: Generating all graph coverings by permutation voltage assignments. Discrete Math. 18, 273–283 (1977)
Gross, J.L., Tucker, T.W.: Topological Graph Theory, Wiley, New York, 1987
Hofmeister, M.: Isomorphisms and automorphisms of coverings. Discrete Math. 98, 175–183 (1991)
Hofmeister, M.: Graph covering projections arising from finite vector spaces over finite fields. Discrete Math. 143, 87–97 (1995)
Hofmeister, M.: A note on counting connected graph covering projections. SIAM J. Disc. Math. 11, 286–292 (1998)
Hong, S., Kwak, J.H., Lee,, J.: Regular graph coverings whose covering transformation groups have the isomorphism extension property. Discrete Math. 148, 85–105 (1996)
Kwak, J.H., Chun, J., Lee, J.: Enumeration of regular graph coverings having finite abelian covering transformation groups. SIAM J. Disc. Math. 11, 273–285 (1998)
Kwak, J.H., Lee, J.: Isomorphism classes of graph bundles. Canad. J. Math. XLII, 747–761 (1990)
Kwak, J.H., Lee, J.: Enumeration of connected graph coverings. J. Graph Theory 23, 105–109 (1996)
Kwak, J.H., Lee, J.: Enumeration of graph coverings, surface branched coverings and related group theory.-In: Hong, S., Kwak, J.H., Kim, K.H., Raush, F.W. (eds.) Combinatorial and Computational Mathematics: Present and Future, World Scientific, Singapore, 2001, pp.97–161
Park, J.H., Chwa, K.Y.: Recursive circulant: a new topology for multicomputer networks. Proc. Internat. Symp. Parallel Architectures, Algorithms and Networks (ISPAN'94), IEEE press, New York, 1994, pp. 73–80
Siagiova, J.: Composition of regular coverings of graphs. J. Electrical Engineering 50, 75–77 (1999)
Siagiova, J.: Composition of regular coverings of graphs and voltage assignments. Australasian J. Combinatorics 28, 131–136 (2003)
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The first author is supported by NSF of China (No. 60473019) and by NKBRPC (2004CB318000), and the second author is supported by Com2MaC-KOSEF (R11-1999-054) in Korea.
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Feng, R., Kwak, J. Circulant Double Coverings of a Circulant Graph of Valency Four. Graphs and Combinatorics 21, 385–400 (2005). https://doi.org/10.1007/s00373-005-0623-2
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DOI: https://doi.org/10.1007/s00373-005-0623-2