97 research outputs found
Edge-disjoint rainbow trees in properly coloured complete graphs
A subgraph of an edge-coloured complete graph is called rainbow if all its edges
have different colours. The study of rainbow decompositions has a long history,
going back to the work of Euler on Latin squares. We discuss three problems
about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth
Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture.
The main result which we discuss is that in every proper edge-colouring of Kn there
are 10−6n edge-disjoint isomorphic spanning rainbow trees. This simultaneously
improves the best known bounds on all these conjectures. Using our method it is also
possible to show that every properly (n−1)-edge-coloured Kn has n/9 edge-disjoint
spanning rainbow trees, giving a further improvement on the Brualdi-Hollingsworth
Conjectur
Multicolored parallelisms of Hamiltonian cycles
AbstractA subgraph in an edge-colored graph is multicolored if all its edges receive distinct colors. In this paper, we prove that a complete graph on 2m+1 vertices K2m+1 can be properly edge-colored with 2m+1 colors in such a way that the edges of K2m+1 can be partitioned into m multicolored Hamiltonian cycles
On the editing distance of graphs
An edge-operation on a graph is defined to be either the deletion of an
existing edge or the addition of a nonexisting edge. Given a family of graphs
, the editing distance from to is the smallest
number of edge-operations needed to modify into a graph from .
In this paper, we fix a graph and consider , the set of
all graphs on vertices that have no induced copy of . We provide bounds
for the maximum over all -vertex graphs of the editing distance from
to , using an invariant we call the {\it binary chromatic
number} of the graph . We give asymptotically tight bounds for that distance
when is self-complementary and exact results for several small graphs
Linearly many rainbow trees in properly edge-coloured complete graphs
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different
colours. The study of rainbow decompositions has a long history, going back to the work of
Euler on Latin squares. In this paper we discuss three problems about decomposing complete
graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and
the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there
are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the
best known bounds on all these conjectures. Using our method we also show that every properly
(n − 1)-edge-coloured Kn has n/9 − 6 edge-disjoint rainbow trees, giving further improvement on
the Brualdi-Hollingsworth Conjectur
Complexity of Token Swapping and its Variants
In the Token Swapping problem we are given a graph with a token placed on
each vertex. Each token has exactly one destination vertex, and we try to move
all the tokens to their destinations, using the minimum number of swaps, i.e.,
operations of exchanging the tokens on two adjacent vertices. As the main
result of this paper, we show that Token Swapping is -hard parameterized
by the length of a shortest sequence of swaps. In fact, we prove that, for
any computable function , it cannot be solved in time where is the number of vertices of the input graph, unless the ETH
fails. This lower bound almost matches the trivial -time algorithm.
We also consider two generalizations of the Token Swapping, namely Colored
Token Swapping (where the tokens have different colors and tokens of the same
color are indistinguishable), and Subset Token Swapping (where each token has a
set of possible destinations). To complement the hardness result, we prove that
even the most general variant, Subset Token Swapping, is FPT in nowhere-dense
graph classes.
Finally, we consider the complexities of all three problems in very
restricted classes of graphs: graphs of bounded treewidth and diameter, stars,
cliques, and paths, trying to identify the borderlines between polynomial and
NP-hard cases.Comment: 23 pages, 7 Figure
Linearly many rainbow trees in properly edge-coloured complete graphs
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different
colours. The study of rainbow decompositions has a long history, going back to the work of
Euler on Latin squares. In this paper we discuss three problems about decomposing complete
graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and
the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there
are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the
best known bounds on all these conjectures. Using our method we also show that every properly
(n − 1)-edge-coloured Kn has n/9 − 6 edge-disjoint rainbow trees, giving further improvement on
the Brualdi-Hollingsworth Conjecture
Decompositions into isomorphic rainbow spanning trees
A subgraph of an edge-coloured graph is called rainbow if all its edges have
distinct colours. Our main result implies that, given any optimal colouring of
a sufficiently large complete graph , there exists a decomposition of
into isomorphic rainbow spanning trees. This settles conjectures of
Brualdi--Hollingsworth (from 1996) and Constantine (from 2002) for large
graphs.Comment: Version accepted to appear in JCT
- …