2,756 research outputs found
Strong chromatic index of sparse graphs
A coloring of the edges of a graph is strong if each color class is an
induced matching of . The strong chromatic index of , denoted by
, is the least number of colors in a strong edge coloring
of . In this note we prove that for every -degenerate graph . This confirms the strong
version of conjecture stated recently by Chang and Narayanan [3]. Our approach
allows also to improve the upper bound from [3] for chordless graphs. We get
that for any chordless graph . Both
bounds remain valid for the list version of the strong edge coloring of these
graphs
Strong chromatic index of k-degenerate graphs
A {\em strong edge coloring} of a graph is a proper edge coloring in
which every color class is an induced matching. The {\em strong chromatic
index} \chiup_{s}'(G) of a graph is the minimum number of colors in a
strong edge coloring of . In this note, we improve a result by D{\k e}bski
\etal [Strong chromatic index of sparse graphs, arXiv:1301.1992v1] and show
that the strong chromatic index of a -degenerate graph is at most
. As a direct consequence, the strong
chromatic index of a -degenerate graph is at most ,
which improves the upper bound by Chang and Narayanan
[Strong chromatic index of 2-degenerate graphs, J. Graph Theory 73 (2013) (2)
119--126]. For a special subclass of -degenerate graphs, we obtain a better
upper bound, namely if is a graph such that all of its -vertices
induce a forest, then \chiup_{s}'(G) \leq 4 \Delta(G) -3; as a corollary,
every minimally -connected graph has strong chromatic index at most . Moreover, all the results in this note are best possible in
some sense.Comment: 3 pages in Discrete Mathematics, 201
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
Strong edge-colouring of sparse planar graphs
A strong edge-colouring of a graph is a proper edge-colouring where each
colour class induces a matching. It is known that every planar graph with
maximum degree has a strong edge-colouring with at most
colours. We show that colours suffice if the graph has girth 6, and
colours suffice if or the girth is at least 5. In the
last part of the paper, we raise some questions related to a long-standing
conjecture of Vizing on proper edge-colouring of planar graphs
Fractional total colourings of graphs of high girth
Reed conjectured that for every epsilon>0 and Delta there exists g such that
the fractional total chromatic number of a graph with maximum degree Delta and
girth at least g is at most Delta+1+epsilon. We prove the conjecture for
Delta=3 and for even Delta>=4 in the following stronger form: For each of these
values of Delta, there exists g such that the fractional total chromatic number
of any graph with maximum degree Delta and girth at least g is equal to
Delta+1
Colouring Graphs with Sparse Neighbourhoods: Bounds and Applications
Let be a graph with chromatic number , maximum degree and
clique number . Reed's conjecture states that for all . It was shown by King and Reed that, provided is large
enough, the conjecture holds for . In this article,
we show that the same statement holds for , thus making
a significant step towards Reed's conjecture. We derive this result from a
general technique to bound the chromatic number of a graph where no vertex has
many edges in its neighbourhood. Our improvements to this method also lead to
improved bounds on the strong chromatic index of general graphs. We prove that
provided is large enough.Comment: Submitted for publication in July 201
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