Faculty of Science
http://hdl.handle.net/1828/64
2021-03-02T05:45:00ZOn the cone of weighted graphs generated by triangles
http://hdl.handle.net/1828/12747
On the cone of weighted graphs generated by triangles
del Valle, Coen; Dukes, Peter J.; Garaschuk, Kseniya
Motivated by problems involving triangle-decompositions of graphs, we examine the facet structure of the cone τn of weighted graphs on n vertices generated by triangles. Our results include enumeration of facets for small n, a construction producing facets of τn+1 from facets of τn, and an arithmetic condition on entries of the normal vectors. We also point out that a copy of τn essentially appears via the perimeter inequalities at one vertex of the metric polytope.
2019-01-01T00:00:00ZLeaves for packings with block size four
http://hdl.handle.net/1828/12746
Leaves for packings with block size four
Chang, Yanxun; Dukes, Peter J.; Feng, Tao
We consider maximum packings of edge-disjoint 4-cliques in the complete graph Kn. When n 1 or 4 (mod 12), these are simply block designs. In other congruence classes, there are necessarily uncovered edges; we examine the possible ‘leave’ graphs induced by those edges. We give particular emphasis to the case n 0 or 3 (mod 12), when the leave is 2-regular. Colbourn
and Ling settled the case of Hamiltonian leaves in this case. We extend their construction and
use several additional direct and recursive constructions to realize a variety of 2-regular leaves.
For various subsets S {3, 4, 5, . . . }, we establish explicit lower bounds on n to guarantee the
existence of maximum packings with any possible leave whose cycle lengths belong to S.
2019-01-01T00:00:00ZLocal balance in graph decompositions
http://hdl.handle.net/1828/12745
Local balance in graph decompositions
Dukes, Peter J.; Bowditch, Flora Caroline
In a balanced graph decomposition, every vertex of the host graph appears in the same number of blocks. We propose the use of colored loops as a framework for unifying various other types of local balance conditions in graph decompositions. In the basic case where a single graph with colored loops is used as a block, an existence theory for such decompositions follows
as a straightforward generalization of previous work on balanced graph decompositions.
2020-01-01T00:00:00ZA lower bound on HMOLS with equal-sized holes
http://hdl.handle.net/1828/12744
A lower bound on HMOLS with equal-sized holes
Dukes, Peter J.; del Valle, Coen; Bailey, Michael
It is known that N(n), the maximum number of mutually orthogonal latin squares
of order n, satisfies the lower bound N(n) > n1/14.8 for large n. For h > 2, relatively little is
known about the quantity N(hn), which denotes the maximum number of ‘HMOLS’ or mutually
orthogonal latin squares having a common equipartition into n holes of a fixed size h. We generalize
a difference matrix method that had been used previously for explicit constructions of HMOLS.
An estimate of R.M. Wilson on higher cyclotomic numbers guarantees our construction succeeds
in suitably large finite fields. Feeding this into a generalized product construction, we are able to
establish the lower bound N(hn) > (log n)1/ for any > 2 and all n > n0(h, ).
2020-01-01T00:00:00Z