Designs for graphs with six vertices and ten edges
Abstract
The design spectrum has been determined for two of the 15 graphs with six vertices and ten edges. In this paper we completely solve the design spectrum problem for a further eight of these graphs.
Keywords
Full Text:
PDFDOI: http://dx.doi.org/10.5614/ejgta.2019.7.2.14
References
P.Adams, E.J. Billington and D.G. Hoffman, On the spectrum for $K_{m+2} K_m$ designs, J. Combin. Des. 5 (1997), 49--60.
P. Adams, D.E. Bryant and M. Buchanan, A survey on the existence of G-designs, J. Combin. Des.16 (2008), 373--410.
P. Adams, D.E. Bryant and A. Khodkar, The spectrum problem for closed $m$-trails, $m le 10$, J. Combin. Math. Combin. Comput. 34 (2000), 223--240.
A.E. Brouwer, A. Schrijver and H. Hanani, Group divisible designs with block size four, Discrete Math. 20 (1977), 1--10.
D.E. Bryant and T.A. McCourt, Existence results for G-designs, http://wiki.smp.uq.edu.au/G-designs.
A.D. Forbes and T.S. Griggs, Designs for graphs with six vertices and nine edges, Australas. J. Combin. 70 (2018), 52--74.
A.D. Forbes, T.S. Griggs and K.A. Forbes, Completing the design spectra for graphs with six vertices and eight edges, Australas. J. Combin. 70 (2018), 386--389.
G. Ge, Group divisible designs, Handbook of Combinatorial Designs, second edition (ed. C.J. Colbourn and J.H. Dinitz), Chapman & Hall/CRC Press (2007), 255--260.
G. Ge, S. Hu, E. Kolotoglu and H. Wei, A Complete Solution to Spectrum Problem for Five-Vertex Graphs with Application to Traffic Grooming in Optical Networks, J. Combin. Des. 23 (2015), 233--273.
G. Ge and A.C.H. Ling, Asymptotic results on the existence of 4-RGDDs and uniform 5-GDDs, J. Combin. Des. 13 (2005), 222--237.
G. Ge and A.C.H. Ling, On the existence of $(K_5 e)$-designs with application to optical networks, SIAM J. Discrete Math. 21 (2008), 851--864.
G. Ge and Y. Miao, PBDs, Frames and Resolvability, Handbook of Combinatorial Designs, second edition (ed. C.J. Colbourn and J.H. Dinitz), Chapman & Hall/CRC Press (2007), 261--265.
R.K. Guy and L.W. Beineke, The coarseness of the complete graph, Canad. J. Math. 20 (1968), 888--894.
H. Hanani, The existence and contruction of balanced incomplete block designs, Ann. Math. Statist. 32 (1961), 361--386.
H. Hanani, D.K. Ray-Chaudhuri and R.M. Wilson, On resolvable designs, Discrete Math. 3 (1972), 343--357.
Q. Kang, H. Zhao and C. Ma, Graph designs for nine graphs with six vertices and nine edges, Ars Combin. 88 (2008), 379--395.
E. Kolotoglu, The Existence and Construction of ($K_5 e $)-Designs of Orders 27, 135, 162, and 216, J. Combin. Des. 21 (2013), 280--302.
B. McKay, Graphs: Simple Graphs: 6 vertices: all (156), http://users.cecs.anu.edu.au/simbdm/data/graphs.html}.
R.C. Mullin, A.L. Poplove and L. Zhu, Decomposition of Steiner triple systems into triangles, J. Combin. Math. Combin. Comp. 1 (1987), 149--174.
R.C. Read and R.J. Wilson, An Atlas of Graphs, Clarendon Press, Oxford, 1998.
Refbacks
- There are currently no refbacks.
ISSN: 2338-2287
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.