Abstract
The cage problem asks for the construction of regular simple graphs with specified degree k, girth g and minimum order n(k; g), (see [5] for a complete survey). This problem was first considered by Tutte [10]. In 1963, Erdös and Sachs [4] proved that (k; g)-cages exist for any given values of k and g.
This research was supported by the Ministry of Education and Science, Spain, and the European Regional Development Fund (ERDF) under project MTM2008-06620-C03-02; and under the Cat-alonian Government project 1298 SGR2009. Partial support by the Spanish MEC (project ARES — CONSOLIDER INGENIO 2010 CSD2007-00004) is acknowledged.
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References
E. Bannai and T. Ito, On finite Moore graphs, J. Fac. Sci. Tokyo, Sect. 1A 20 (1973), 191–208.
N. Biggs and T. Ito, Graphs with even girth and small excess, Math. Proc. Cambridge Philos. Soc. 88 (1980) 1–10.
R. M. Damerell, On Moore graphs, Proc. Cambridge Phil. Soc. 74 (1973), 227–236.
P. Erdős and H. Sachs, Regulare Graphen gegebener Taillenweite mit minimaler Knotenzahl, Wiss. Z. Uni. Halle (Math. Nat.) 12 (1963), 251–257.
G. Exoo and R. Jajcay, Dynamic Cage Survey, Electron. J. Combin. 15(2008), #DS16.
F. Harary and P. Kovács, Regular graphs with given girth pair, J. Graph Theory 7 (1983), 209–218.
P. Kovács, The minimal trivalent graphs with given smallest odd cycle, Discrete Math. 54 (1985), 295–299.
C. M. Campbell, On cages for girth pair (6, b), Discrete Math. 177 (1997), 259–266.
T. Pisanski, M. Boben, D. Marusic, A. Orbanic and A. Graovac, The 10-cages and derived configurations, Discrete Math. 275 (2004), 265–276.
W. T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. (1947), 459–474.
B.-G. Xu, P. Wang and J.-f. Wang, On the Monotonicity of (k; g, h)-graphs, Acta Mathematicae Applicatae Sinica, English Series 18(3) (2002), 477–480.
P. K. Wong, Cages-a survey, J. Graph Theory 6 (1982), 1–22.
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Salas, J., Balbuena, C. (2013). On the order of cages with a given girth pair. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_83
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DOI: https://doi.org/10.1007/978-88-7642-475-5_83
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