40 research outputs found
The Hamilton-Waterloo Problem with even cycle lengths
The Hamilton-Waterloo Problem HWP asks for a
2-factorization of the complete graph or , the complete graph with
the edges of a 1-factor removed, into -factors and
-factors, where . In the case that and are both
even, the problem has been solved except possibly when
or when and are both odd, in which case necessarily . In this paper, we develop a new construction that creates
factorizations with larger cycles from existing factorizations under certain
conditions. This construction enables us to show that there is a solution to
HWP for odd and whenever the obvious
necessary conditions hold, except possibly if ; and
; ; or . This result almost completely
settles the existence problem for even cycles, other than the possible
exceptions noted above
On non-isomorphic biminimal pots realizing the cube
In this paper, we disprove a conjecture recently proposed in [L. Almodovar et
al., arXiv:2108.00035] on the non-existence of biminimal pots realizing the
cube, namely pots with the minimum number of tiles and the minimum number of
bond-edge types. In particular, we present two biminimal pots realizing the
cube and show that these two pots are unique up to isomorphisms
Constructing uniform 2-factorizations via row-sum matrices: solutions to the Hamilton-Waterloo problem
In this paper, we formally introduce the concept of a row-sum matrix over an
arbitrary group . When is cyclic, these types of matrices have been
widely used to build uniform 2-factorizations of small Cayley graphs (or,
Cayley subgraphs of blown-up cycles), which themselves factorize complete
(equipartite) graphs.
Here, we construct row-sum matrices over a class of non-abelian groups, the
generalized dihedral groups, and we use them to construct uniform
-factorizations that solve infinitely many open cases of the
Hamilton-Waterloo problem, thus filling up large parts of the gaps in the
spectrum of orders for which such factorizations are known to exist
2-Starters, Graceful Labelings, and a Doubling Construction for the Oberwolfach Problem
Every 1-rotational solution of a classic or twofold Oberwolfach problem (OP) of order n is generated by a suitable 2-factor (starter) of or , respectively. It is shown that any starter of a twofold OP of order n gives rise to a starter of a classic OP of order 2n-1 (doubling construction). It is also shown that by suitably modifying the starter of a classic OP, one may obtain starters of some other OPs of the same order but having different parameters. The combination of these two constructions leads to lots of new infinite classes of solvable OPs. Still more classes can be obtained with the help of a third construction making use of the possible gracefulness of a graph whose connected components are cycles and at most one path. As one of the many applications, Hilton and Johnson's [J London Math Soc, 64 (2001) 513â522] bound about the solvability of OP(r,s) is improved to in the case of r even
Factorizing the Rado graph and infinite complete graphs
Let F = {Fα : α â A} be a family of infinite graphs, together with Î. The Factorization Problem FP(F, Î) asks whether F can be realized as a factorization of Î, namely, whether there is a factorization G = {Îα : α â A} of Î such that each Îα is a copy of Fα. We study this problem when Î is either the Rado graph R or the complete graph Kâ” of infinite order â”. When F is a countably infinite family, we show that FP(F, R) is solvable if and only if each graph in F has no finite dominating set. We also prove that FP(F, Kâ”) admits a solution whenever the cardinality of F coincides with the order and the domination numbers of its graphs. For countable complete graphs, we show some non existence results when the domination numbers of the graphs in F are finite. More precisely, we show that there is no factorization of KN into copies of a k-star (that is, the vertex disjoint union of k countable stars) when k = 1, 2, whereas it exists when k â„ 4, leaving the problem open for k = 3. Finally, we determine sufficient conditions for the graphs of a decomposition to be arranged into resolution classes