71 research outputs found
Rainbow Perfect and Near-Perfect Matchings in Complete Graphs with Edges Colored by Circular Distance
Given an edge-colored complete graph Kn on n vertices, a perfect (respectively, near-perfect) matching M in Kn with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we consider an edge coloring of Kn by circular distance, and we denote the resulting complete graph by K●n. We show that when K●n has an even number of vertices, it contains a rainbow perfect matching if and only if n=8k or n=8k+2, where k is a nonnegative integer. In the case of an odd number of vertices, Kirkman matching is known to be a rainbow near-perfect matching in K●n. However, real-world applications sometimes require multiple rainbow near-perfect matchings. We propose a method for using a recursive algorithm to generate multiple rainbow near-perfect matchings in K●n
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
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow independent sets on dense graph classes
Given a family of independent sets in a graph, a rainbow
independent set is an independent set such that there is an injection
where for each , is contained in
. Aharoni, Briggs, J. Kim, and M. Kim [Rainbow independent sets in
certain classes of graphs. arXiv:1909.13143] determined for various graph
classes whether satisfies a property that for every
, there exists such that every family of
independent sets of size in a graph in contains a rainbow
independent set of size . In this paper, we add two dense graph classes
satisfying this property, namely, the class of graphs of bounded neighborhood
diversity and the class of -powers of graphs in a bounded expansion class
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
Graph tilings in incompatibility systems
Given two graphs and , an \emph{-tiling} of is a collection of
vertex-disjoint copies of in and an \emph{-factor} is an -tiling
that covers all vertices of . K\"{u}hn and Osthus managed to characterize,
up to an additive constant, the minimum degree threshold which forces an
-factor in a host graph . In this paper we study a similar tiling problem
in a system that is locally bounded. An \emph{incompatibility system}
over is a family with
. We say that two
edges are \emph{incompatible} if for some
, and otherwise \emph{compatible}. A subgraph of is
\emph{compatible} if every pair of edges in are compatible. An
incompatibility system is \emph{-bounded} if for any
vertex and any edge incident with , there are at most
two-subsets in containing . This notion was partly motivated by a
concept of transition system introduced by Kotzig in 1968, and first formulated
by Krivelevich, Lee and Sudakov to study the robustness of Hamiltonicity of
Dirac graphs.
We prove that for any and any graph with vertices, there
exists a constant such that for any sufficiently large with , if is an -vertex graph with
and is a -bounded incompatibility system over , then there exists a compatible
-factor in , where the value is either the chromatic number
or the critical chromatic number and we provide a
dichotomy. Moreover, the error term is inevitable in general case
- …