8 research outputs found
On rainbow tetrahedra in Cayley graphs
Let be the complete undirected Cayley graph of the odd cyclic
group . Connected graphs whose vertices are rainbow tetrahedra in
are studied, with any two such vertices adjacent if and only if they
share (as tetrahedra) precisely two distinct triangles. This yields graphs
of largest degree 6, asymptotic diameter and almost all vertices
with degree: {\bf(a)} 6 in ; {\bf(b)} 4 in exactly six connected subgraphs
of the -semi-regular tessellation; and {\bf(c)} 3 in exactly four
connected subgraphs of the -regular hexagonal tessellation. These
vertices have as closed neighborhoods the union (in a fixed way) of closed
neighborhoods in the ten respective resulting tessellations. Generalizing
asymptotic results are discussed as well.Comment: 21 pages, 7 figure
Transversals in generalized Latin squares
We are seeking a sufficient condition that forces a transversal in a
generalized Latin square. A generalized Latin square of order is equivalent
to a proper edge-coloring of . A transversal corresponds to a
multicolored perfect matching. Akbari and Alipour defined as the least
integer such that every properly edge-colored , which contains at
least different colors, admits a multicolored perfect matching. They
conjectured that if is large enough. In this note we prove
that is bounded from above by if . We point out a
connection to anti-Ramsey problems. We propose a conjecture related to a
well-known result by Woolbright and Fu, that every proper edge-coloring of
admits a multicolored -factor
Bounded degree graphs and hypergraphs with no full rainbow matchings
Given a multi-hypergraph that is edge-colored into color classes , a full rainbow matching is a matching of that contains
exactly one edge from each color class . One way to guarantee the
existence of a full rainbow matching is to have the size of each color class
be sufficiently large compared to the maximum degree of . In this
paper, we apply a simple iterative method to construct edge-colored
multi-hypergraphs with a given maximum degree, large color classes, and no full
rainbow matchings. First, for every and , we construct
edge-colored -uniform multi-hypergraphs with maximum degree such
that each color class has size and there is no full
rainbow matching, which demonstrates that a theorem of Aharoni, Berger, and
Meshulam (2005) is best possible. Second, we construct properly edge-colored
multigraphs with no full rainbow matchings which disprove conjectures of
Delcourt and Postle (2022). Finally, we apply results on full rainbow matchings
to list edge-colorings and prove that a color degree generalization of Galvin's
theorem (1995) does not hold