8 research outputs found
On Hamilton decompositions of infinite circulant graphs
The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph).
Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is Β±{1,2,...,k} or Β±{1,2,...,k - 1, 1,2,...,k + 1}
Perfect 1-factorisations of circulants with small degree
A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (perfect matchings), and a perfect 1-factorisation is a 1-factorisation in which the union of any two of the 1-factors is a Hamilton cycle. We consider the problem of the existence of perfect 1-factorisations of even order circulant graphs with small degree. In particular, we characterise the 3-regular circulant graphs that admit a perfect 1-factorisation and we solve the existence problem for a large family of 4-regular circulants. Results of computer searches for perfect 1-factorisations of 4-regular circulant graphs of orders up to 30 are provided and some problems are posed
On factorisations of complete graphs into circulant graphs and the Oberwolfach problem
Various results on factorisations of complete graphs into circulant graphs and on 2-factorisations of these circulant graphs are proved. As a consequence, a number of new results on the Oberwolfach Problem are obtained. For example, a complete solution to the Oberwolfach Problem is given for every 2-regular graph of order 2p where p β‘ 5 (mod 8) is prime
Some results on decompositions of low degree circulant graphs
The circulant graph of order n with connection set S is denoted by Circ(n, S). Several results on decompositions of Circ(n, {1, 2}) and Circ(n, {1, 2, 3}) are proved here. The existence problems for decompositions into paths of arbitrary specified lengths and for decompositions into cycles of arbitrary specified lengths are completely solved for Circ(n, {1, 2}). For all m 3 and m + m + β’ β’ β’ + m = 3n. This settles the problem of decomposing Circ(n, {1, 2, 3}) into specified numbers of 3-cycles, 4-cycles and 5-cycles
Decompositions of complete graphs into cycles of arbitrary lengths
We show that the complete graph on vertices can be decomposed into
cycles of specified lengths if and only if is odd, for , and . We also show
that the complete graph on vertices can be decomposed into a perfect
matching and cycles of specified lengths if and only if
is even, for , and .Comment: 182 pages, 0 figures, A condensed version of this paper was published
as "Cycle decompositions V: Complete graphs into cycles of arbitrary lengths"
(see reference [24]). Here, we include supplementary data and some proofs
which were omitted from that pape