7 research outputs found
Distance edge-colourings and matchings
AbstractWe consider a distance generalisation of the strong chromatic index and the maximum induced matching number. We study graphs of bounded maximum degree and Erdős–Rényi random graphs. We work in three settings. The first is that of a distance generalisation of an Erdős–Nešetřil problem. The second is that of an upper bound on the size of a largest distance matching in a random graph. The third is that of an upper bound on the distance chromatic index for sparse random graphs. One of our results gives a counterexample to a conjecture of Skupień
The distance-t chromatic index of graphs
We consider two graph colouring problems in which edges at distance at most
are given distinct colours, for some fixed positive integer . We obtain
two upper bounds for the distance- chromatic index, the least number of
colours necessary for such a colouring. One is a bound of (2-\eps)\Delta^t
for graphs of maximum degree at most , where \eps is some absolute
positive constant independent of . The other is a bound of (as ) for graphs of maximum degree at most
and girth at least . The first bound is an analogue of Molloy and Reed's
bound on the strong chromatic index. The second bound is tight up to a constant
multiplicative factor, as certified by a class of graphs of girth at least ,
for every fixed , of arbitrarily large maximum degree , with
distance- chromatic index at least .Comment: 14 pages, 2 figures; to appear in Combinatorics, Probability and
Computin
Unified almost linear kernels for generalized covering and packing problems on nowhere dense classes
Let be a family of graphs, and let be nonnegative
integers. The \textsc{-Covering} problem asks whether for a
graph and an integer , there exists a set of at most vertices in
such that has no induced subgraph isomorphic to a
graph in , where is the -th power of . The
\textsc{-Packing} problem asks whether for a graph and
an integer , has induced subgraphs such that each
is isomorphic to a graph in , and for distinct , the distance between and in is larger than
.
We show that for every fixed nonnegative integers and every fixed
nonempty finite family of connected graphs, the
\textsc{-Covering} problem with and the
\textsc{-Packing} problem with
admit almost linear kernels on every nowhere dense class of graphs, and admit
linear kernels on every class of graphs with bounded expansion, parameterized
by the solution size . We obtain the same kernels for their annotated
variants. As corollaries, we prove that \textsc{Distance- Vertex Cover},
\textsc{Distance- Matching}, \textsc{-Free Vertex Deletion},
and \textsc{Induced--Packing} for any fixed finite family
of connected graphs admit almost linear kernels on every nowhere
dense class of graphs and linear kernels on every class of graphs with bounded
expansion. Our results extend the results for \textsc{Distance- Dominating
Set} by Drange et al. (STACS 2016) and Eickmeyer et al. (ICALP 2017), and the
result for \textsc{Distance- Independent Set} by Pilipczuk and Siebertz (EJC
2021).Comment: 38 page