176 research outputs found
Double domination and total -domination in digraphs and their dual problems
A subset of vertices of a digraph is a double dominating set (total
-dominating set) if every vertex not in is adjacent from at least two
vertices in , and every vertex in is adjacent from at least one vertex
in (the subdigraph induced by has no isolated vertices). The double
domination number (total -domination number) of a digraph is the minimum
cardinality of a double dominating set (total -dominating set) in . In
this work, we investigate these concepts which can be considered as two
extensions of double domination in graphs to digraphs, along with the concepts
-limited packing and total -limited packing which have close
relationships with the above-mentioned concepts
Hamilton decompositions of regular expanders: applications
In a recent paper, we showed that every sufficiently large regular digraph G
on n vertices whose degree is linear in n and which is a robust outexpander has
a decomposition into edge-disjoint Hamilton cycles. The main consequence of
this theorem is that every regular tournament on n vertices can be decomposed
into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large.
This verified a conjecture of Kelly from 1968. In this paper, we derive a
number of further consequences of our result on robust outexpanders, the main
ones are the following: (i) an undirected analogue of our result on robust
outexpanders; (ii) best possible bounds on the size of an optimal packing of
edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range
of values for d. (iii) a similar result for digraphs of given minimum
semidegree; (iv) an approximate version of a conjecture of Nash-Williams on
Hamilton decompositions of dense regular graphs; (v) the observation that dense
quasi-random graphs are robust outexpanders; (vi) a verification of the `very
dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint
Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the
size of an optimal packing of edge-disjoint Hamilton cycles in a random
tournament.Comment: final version, to appear in J. Combinatorial Theory
Orientable domination in product-like graphs
The orientable domination number, , of a graph is the
largest domination number over all orientations of . In this paper, is studied on different product graphs and related graph operations. The
orientable domination number of arbitrary corona products is determined, while
sharp lower and upper bounds are proved for Cartesian and lexicographic
products. A result of Chartrand et al. from 1996 is extended by establishing
the values of for arbitrary positive integers
and . While considering the orientable domination number of
lexicographic product graphs, we answer in the negative a question concerning
domination and packing numbers in acyclic digraphs posed in [Domination in
digraphs and their direct and Cartesian products, J. Graph Theory 99 (2022)
359-377]
Approximate Hamilton decompositions of robustly expanding regular digraphs
We show that every sufficiently large r-regular digraph G which has linear
degree and is a robust outexpander has an approximate decomposition into
edge-disjoint Hamilton cycles, i.e. G contains a set of r-o(r) edge-disjoint
Hamilton cycles. Here G is a robust outexpander if for every set S which is not
too small and not too large, the `robust' outneighbourhood of S is a little
larger than S. This generalises a result of K\"uhn, Osthus and Treglown on
approximate Hamilton decompositions of dense regular oriented graphs. It also
generalises a result of Frieze and Krivelevich on approximate Hamilton
decompositions of quasirandom (di)graphs. In turn, our result is used as a tool
by K\"uhn and Osthus to prove that any sufficiently large r-regular digraph G
which has linear degree and is a robust outexpander even has a Hamilton
decomposition.Comment: Final version, published in SIAM Journal Discrete Mathematics. 44
pages, 2 figure
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