42 research outputs found
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
Among graphs, groups, and latin squares
A latin square of order n is an n × n array in which each row and each column contains each of the numbers {1, 2, . . . , n}. A k-plex in a latin square is a collection of entries which intersects each row and column k times and contains k copies of each symbol. This thesis studies the existence of k-plexes and approximations of k-plexes in latin squares, paying particular attention to latin squares which correspond to multiplication tables of groups. The most commonly studied class of k-plex is the 1-plex, better known as a transversal. Although many latin squares do not have transversals, Brualdi conjectured that every latin square has a near transversal—i.e. a collection of entries with distinct symbols which in- tersects all but one row and all but one column. Our first main result confirms Brualdi’s conjecture in the special case of group-based latin squares. Then, using a well-known equivalence between edge-colorings of complete bipartite graphs and latin squares, we introduce Hamilton 2-plexes. We conjecture that every latin square of order n ≥ 5 has a Hamilton 2-plex and provide a range of evidence for this conjecture. In particular, we confirm our conjecture computationally for n ≤ 8 and show that a suitable analogue of Hamilton 2-plexes always occur in n × n arrays with no symbol appearing more than n/√96 times. To study Hamilton 2-plexes in group-based latin squares, we generalize the notion of harmonious groups to what we call H2-harmonious groups. Our second main result classifies all H2-harmonious abelian groups. The last part of the thesis formalizes an idea which first appeared in a paper of Cameron and Wanless: a (k,l)-plex is a collection of entries which intersects each row and column k times and contains at most l copies of each symbol. We demonstrate the existence of (k, 4k)-plexes in all latin squares and (k, k + 1)-plexes in sufficiently large latin squares. We also find analogues of these theorems for Hamilton 2-plexes, including our third main result: every sufficiently large latin square has a Hamilton (2,3)-plex