Provider
Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije
(Obvezni izvod spletne publikacije)
description
The generalized Oberwolfach problem ?$\rm{OP}_t(2w + 1; N_1, N_2, \dots , Nt; \alpha_1, \alpha_2, \dots, \alpha_t)$? asks for a factorization of ?$K_{2w + 1}$? into ?$\alpha_i C_{N_i}$?-factors (where a ?$C_{N_i}$?-factor of ?$K_{2w + 1}$? is a spanning subgraph whose components are cycles of length ?$N_i \ge 3$?) for ?$i=1, 2, \dots , t$?. Necessarily, ?$N=\mathrm{lcm}(N_1, N_2, \dots, N_t)$? is a divisor of ?$2w + 1$? and ?$w=\sum^t_{i=1} \alpha_i$?. For ?$t=1$? we have the classic Oberwolfach problem. For ?$t=2$? this is the well-studied Hamilton-Waterloo problem, whereas for ?$t \ge 3$? very little is known. In this paper, we show, among other things, that the above necessary conditions are sufficient whenever ?$2w + 1 \ge (t + 1)N$?, ?$\alpha_i > 1$? for every ?$i \in \{1, 2, \dots, t\}$?, and ?$\mathrm{gcd} (N_1, N_2, \dots, N_t) > 1$?. We also provide sufficient conditions for the solvability of the generalized Oberwolfach problem over an arbitrary graph and, in particular, the complete equipartite graph.