### On z-cycle factorizations with two associate classes where z is 2a and a is even

#### Abstract

Let *K* = *K*(*a*, *p*; *λ*_{1}, *λ*_{2}) be the multigraph with: the number of parts equal to *p*; the number of vertices in each part equal to *a*; the number of edges joining any two vertices of the same part equal to *λ*_{1}; and the number of edges joining any two vertices of different parts equal to *λ*_{2}. The existence of *C*_{4}-factorizations of *K* has been settled when *a* is even; when *a* ≡ 1 (mod 4) with one exception; and for very few cases when *a* ≡ 3 (mod 4). The existence of *C*_{z}-factorizations of *K* has been settled when *a* ≡ 1 (mod *z*) and *λ*_{1} is even, and when *a* ≡ 0 (mod *z*). In this paper, we give a construction for *C*_{z}-factorizations of *K* for *z* = 2*a* when *a* is even.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2023.11.2.9

#### References

R.C. Bose and T. Shimamoto, Classification and analysis of partially balanced incomplete block designs with two associate classes, Journal of the American Statistical Association, 47 (1952), 151–184.

E.J. Billington and C.A. Rodger, Resolvable 4-cycle group divisible designs with two associate classes: part size even, Discrete Math, 308 (2008), 303–307.

H.L. Fu, C.A. Rodger, and D.G. Sarvate, The existence of group divisible designs with first and second associates, having block size 3, Ars Combin., 54 (2000), 33–50.

H.L. Fu and C.A. Rodger, 4-cycle group-divisible designs with two associate classes, Combin. Probab. Comput., 10 (2001), 317–343.

H.L. Fu and C.A. Rodger, Group divisible designs with two associate classes: n = 2 or m = 2, J. Combin. Theory (A), 83 (1998), 94–117.

C. Goss, M. Tiemeyer, and R. Waller, On C4-factorizations with two associate classes where a ≡ 3 (mod 4) and small, Congr. Numer. 219 (2014), 97–128; MR3308539.

J. Liu, A generalization of the Oberwolfach problem and Ct-factorizations of complete equipartite grpahs, J. Combin. Designs, 8 (2000), 42–49.

C.A. Rodger and M.A. Tiemeyer, C4-Factorizations with Two Associate Classes, Australas. J. Combin. 40 (2008), 217–228.

C.A. Rodger and M.A.Tiemeyer, C4-factorizations with two associate classes, λ1 is odd, Australas. J. Combin. 50 (2011), 259–278.

M.A. Tiemeyer, On z-cycle factorizations with two associate classes where z is even and a is 0 or 1 mod z, J. Combin. Math. and Comb. Computing, 105 (2018), 11–19.

### Refbacks

- There are currently no refbacks.

ISSN: 2338-2287

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.