Abstract
In this paper, the fine triangle intersection problem for a pair of maximum kite packings is investigated. Let Fin(v) = {(s, t): ∃ a pair of maximum kite packings of order v intersecting in s blocks and s + t triangles}. Let Adm(v) = {(s, t): s + t ≤ b v , s, t are non-negative integers}, where b v = └v (v −1)/8┘. It is established that Fin(v) = Adm(v)\{(b v − 1, 0), (b v − 1, 1)} for any integer v ≡ 0,1 (mod 8) and v ≥ 8; Fin(v) = Adm(v) for any integer v ≡ 2, 3, 4, 5, 6,7 (mod 8) and v ≥ 4.
Similar content being viewed by others
References
Kramer, E. S., Mesner, D. M.: Intersections among Steiner systems. J. Combin. Theory Ser. A, 16, 273–285 (1974)
Lindner, C. C., Rosa, A.: Steiner triple systems having a prescribed number of triples in common. Canad. J. Math., 27, 1166–1175 (1975). Corrigendum: Canad. J. Math., 30, 896 (1978)
Colbourn, C. J., Hoffman, D. G., Lindner, C. C.: Intersections of S (2, 4, v) designs. Ars Combin., 33, 97–111 (1992)
Billington, E. J., Kreher, D. L.: The intersection problem for small G-designs. Australas. J. Combin., 12, 239–258 (1995)
Billington, E. J.: The intersection problem for combinatorial designs. Congr. Numer., 92, 33–54 (1993)
Billington, E. J., Gionfriddo, M., Lindner, C. C.: The intersection problem for K 4 − e designs. J. Statist. Plann. Inference, 58, 5–27 (1997)
Butler, R. A. R., Hoffman, D. G.: Intersections of group divisible triple systems. Ars Combin., 34, 268–288 (1992)
Chang, Y., Lo Faro, G.: Intersection numbers of Kirkman triple systems. J. Combin. Theory Ser. A, 86, 348–361 (1999)
Chang, Y., Lo Faro, G.: Intersection numbers of Latin squares with their own orthogonal mates. Australas. J. Combin., 26, 283–304 (2002)
Fu, H. L.: On the construction of certain types of latin squares with prescribed intersections, Ph.D. Thesis, Auburn University, Alabama, 1980
Gionfriddo, M., Lindner, C. C.: Construction of Steiner quadruple systems having a prescribed number of blocks in common. Discrete Math., 34, 31–42 (1981)
Hoffman, D. G., Lindner, C. C.: The flower intersection problem for Steiner triple systems. Ann. Discrete Math., 34, 243–258 (1987)
Lindner, C. C., Yazici, E. S.: The triangle intersection problem for kite systems. Ars Combin., 75, 225–231 (2005)
Billington, E. J., Yazici, E. S., Lindner, C. C.: The triangle intersection problem for K 4−e designs. Utilitas Math., 73, 3–21 (2007)
Chang, Y., Feng, T., Lo Faro, G.: The triangle intersection problem for S(2, 4, v) designs. Discrete Math., 310, 3194–3205 (2010)
Chang, Y., Feng, T., Lo Faro, G., et al.: The fine triangle intersection problem for kite systems. Discrete Math., 312, 545–553 (2012)
Chang, Y., Feng, T., Lo Faro, G., et al.: Enumerations of ( K 4 −e)-designs with small orders. Quaderni di Matematica (special volume dedicated to the memory of Lucia Gionfriddo), in press
Chang, Y., Feng, T., Lo Faro, G., et al.: The fine triangle intersection problem for (K 4 −e)-designs. Discrete Math., 311, 2442–2462 (2011)
Chang, Y., Lo Faro, G., Tripodi, A.: Tight blocking sets in some maximum packings of λK n. Discrete Math., 308, 427–438 (2008)
Wilson, R. M.: Constructions and uses of pairwise balanced designs. Math. Centre Tracts, 55, 18–41 (1974)
Colbourn, C. J., Hoffman, D. G., Rees, R.: A new class of group divisible designs with blocks size three. J. Combin. Theory Ser. A, 59, 73–89 (1992)
Zhang, G., Chang, Y., Feng, T.: The fine triangle intersections for maximum kite packings. ArXiv:1207.3931
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the Fundamental Research Funds for the Central Universities (Grant Nos. 2011JBZ012 and 2011JBM298) and National Natural Science Foundation of China (Grant Nos. 61071221 and 10901016)
Electronic supplementary material
Rights and permissions
About this article
Cite this article
Zhang, G.Z., Chang, Y.X. & Feng, T. The fine triangle intersections for maximum kite packings. Acta. Math. Sin.-English Ser. 29, 867–882 (2013). https://doi.org/10.1007/s10114-013-1736-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-013-1736-9