Theta graphs
From Gdesigns
Relevant articles: _{[1]}, _{[2]}, _{[3]}, _{[4]}, _{[5]}, _{[6]}, _{[7]}, _{[8]}.
Theta graphs
The theta graph is the graph consisting of three internally disjoint paths with common endpoints and lengths and with .
Spectrum Results
Table 1
summarises the known results on the spectrum problem for theta graphs
with up to nine edges. An explanation of the sources of these results is
given in _{[1]}.
Table 1
Spectrum for theta graphs with edges  Exceptions  

 
 

Theorem 1 _{[4]}
There exists a design of order in each of the following cases.
 is odd and except when .
 and .
 and .
 , and .
Theorem 2 _{[6]}, _{[7]}, _{[8]}
Let , let and let . There exists a design of order .
References
 ↑ ^{1.0} ^{1.1} Adams, P., Bryant, D., and Buchanan, M. A survey on the existence of Gdesigns, J. Combin. Des. 16, 373–410 (2008).
 ↑ Bermond, J. C., Huang, C., Rosa, A., and Sotteau, D. Decomposition of complete graphs into isomorphic subgraphs with five vertices, Ars Combin. 10, 211–254 (1980).
 ↑ Bermond, J. C. and Schönheim, J. $Gdecomposition of K_n, where G has four vertices or less, Discrete Math. 19, 113–120 (1977).
 ↑ ^{4.0} ^{4.1} Blinco, A. On diagonal cycle systems, Australas. J. Combin. 24, 221–230 (2001).
 ↑ Blinco, A. Decompositions of complete graphs into theta graphs with fewer than ten edges, Util. Math. 64, 197–212 (2003).
 ↑ ^{6.0} ^{6.1} Delorme, C., Maheo, M., Thuillier, H., Koh, K. M., and Teo, H. K. Cycles with a chord are graceful, J. Graph Theory, 4, 409–415 (1980).
 ↑ ^{7.0} ^{7.1} Koh, K. M. and Yap, K. Y. Graceful numberings of cycles with a P_3chord, Bull. Inst. Math. Acad. Sinica, 13, 41–48 (1985).
 ↑ ^{8.0} ^{8.1} Punnim, N. and Pabhapote, N. On graceful graphs: cycles with a P_kchord, k\geq 4, Ars Combin. 23, 225–228 (1987).