1,670 research outputs found
Decompositions of complete graphs into bipartite 2-regular subgraphs
It is shown that if G is any bipartite 2-regular graph of order at most n/2 or at least n – 2, then the obvious necessary conditions are sufficient for the existence of a decomposition of the complete graph of order n into a perfect matching and edge-disjoint copies of G
Hamilton decompositions of regular tournaments
We show that every sufficiently large regular tournament can almost
completely be decomposed into edge-disjoint Hamilton cycles. More precisely,
for each \eta>0 every regular tournament G of sufficiently large order n
contains at least (1/2-\eta)n edge-disjoint Hamilton cycles. This gives an
approximate solution to a conjecture of Kelly from 1968. Our result also
extends to almost regular tournaments.Comment: 38 pages, 2 figures. Added section sketching how we can extend our
main result. To appear in the Proceedings of the LM
Decomposing Cubic Graphs into Connected Subgraphs of Size Three
Let be the set of connected graphs of size 3. We
study the problem of partitioning the edge set of a graph into graphs taken
from any non-empty . The problem is known to be NP-complete for
any possible choice of in general graphs. In this paper, we assume that
the input graph is cubic, and study the computational complexity of the problem
of partitioning its edge set for any choice of . We identify all polynomial
and NP-complete problems in that setting, and give graph-theoretic
characterisations of -decomposable cubic graphs in some cases.Comment: to appear in the proceedings of COCOON 201
Rainbow Hamilton cycles in random regular graphs
A rainbow subgraph of an edge-coloured graph has all edges of distinct
colours. A random d-regular graph with d even, and having edges coloured
randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with
probability tending to 1 as n tends to infinity, provided d is at least 8.Comment: 16 page
Decompositions into subgraphs of small diameter
We investigate decompositions of a graph into a small number of low diameter
subgraphs. Let P(n,\epsilon,d) be the smallest k such that every graph G=(V,E)
on n vertices has an edge partition E=E_0 \cup E_1 \cup ... \cup E_k such that
|E_0| \leq \epsilon n^2 and for all 1 \leq i \leq k the diameter of the
subgraph spanned by E_i is at most d. Using Szemer\'edi's regularity lemma,
Polcyn and Ruci\'nski showed that P(n,\epsilon,4) is bounded above by a
constant depending only \epsilon. This shows that every dense graph can be
partitioned into a small number of ``small worlds'' provided that few edges can
be ignored. Improving on their result, we determine P(n,\epsilon,d) within an
absolute constant factor, showing that P(n,\epsilon,2) = \Theta(n) is unbounded
for \epsilon
n^{-1/2} and P(n,\epsilon,4) = \Theta(1/\epsilon) for \epsilon > n^{-1}. We
also prove that if G has large minimum degree, all the edges of G can be
covered by a small number of low diameter subgraphs. Finally, we extend some of
these results to hypergraphs, improving earlier work of Polcyn, R\"odl,
Ruci\'nski, and Szemer\'edi.Comment: 18 page
On the decomposition threshold of a given graph
We study the -decomposition threshold for a given graph .
Here an -decomposition of a graph is a collection of edge-disjoint
copies of in which together cover every edge of . (Such an
-decomposition can only exist if is -divisible, i.e. if and each vertex degree of can be expressed as a linear combination of
the vertex degrees of .)
The -decomposition threshold is the smallest value ensuring
that an -divisible graph on vertices with
has an -decomposition. Our main results imply
the following for a given graph , where is the fractional
version of and :
(i) ;
(ii) if , then
;
(iii) we determine if is bipartite.
In particular, (i) implies that . Our proof
involves further developments of the recent `iterative' absorbing approach.Comment: Final version, to appear in the Journal of Combinatorial Theory,
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