9,003 research outputs found
Sequence variations of the 1-2-3 Conjecture and irregularity strength
Karonski, Luczak, and Thomason (2004) conjectured that, for any connected
graph G on at least three vertices, there exists an edge weighting from {1,2,3}
such that adjacent vertices receive different sums of incident edge weights.
Bartnicki, Grytczuk, and Niwcyk (2009) made a stronger conjecture, that each
edge's weight may be chosen from an arbitrary list of size 3 rather than
{1,2,3}. We examine a variation of these conjectures, where each vertex is
coloured with a sequence of edge weights. Such a colouring relies on an
ordering of the graph's edges, and so two variations arise -- one where we may
choose any ordering of the edges and one where the ordering is fixed. In the
former case, we bound the list size required for any graph. In the latter, we
obtain a bound on list sizes for graphs with sufficiently large minimum degree.
We also extend our methods to a list variation of irregularity strength, where
each vertex receives a distinct sequence of edge weights.Comment: Accepted to Discrete Mathematics and Theoretical Computer Scienc
Distant irregularity strength of graphs with bounded minimum degree
Consider a graph without isolated edges and with maximum degree
. Given a colouring , the weighted degree of a
vertex is the sum of its incident colours, i.e., .
For any integer , the least admitting the existence of such
attributing distinct weighted degrees to any two different vertices at distance
at most in is called the -distant irregularity strength of and
denoted by . This graph invariant provides a natural link between the
well known 1--2--3 Conjecture and irregularity strength of graphs. In this
paper we apply the probabilistic method in order to prove an upper bound
for graphs with minimum degree , improving thus far best upper bound .Comment: 11 page
Distant sum distinguishing index of graphs
Consider a positive integer and a graph with maximum degree
and without isolated edges. The least so that a proper edge
colouring exists such that for every pair of distinct vertices at distance at
most in is denoted by . For it has been
proved that . For any in turn an
infinite family of graphs is known with
. We prove that on the other hand,
for . In particular we show that
if .Comment: 10 page
Distant total irregularity strength of graphs via random vertex ordering
Let be a (not necessarily proper) total
colouring of a graph with maximum degree . Two vertices
are sum distinguished if they differ with respect to sums of their
incident colours, i.e. . The
least integer admitting such colouring under which every at
distance in are sum distinguished is denoted by . Such graph invariants link the concept of the total vertex
irregularity strength of graphs with so called 1-2-Conjecture, whose concern is
the case of . Within this paper we combine probabilistic approach with
purely combinatorial one in order to prove that for every integer and each graph , thus
improving the previously best result: .Comment: 8 page
The 1-2-3 Conjecture and related problems: a survey
The 1-2-3 Conjecture, posed in 2004 by Karonski, Luczak, and Thomason, is as
follows: "If G is a graph with no connected component having exactly 2
vertices, then the edges of G may be assigned weights from the set {1,2,3} so
that, for any adjacent vertices u and v, the sum of weights of edges incident
to u differs from the sum of weights of edges incident to v." This survey paper
presents the current state of research on the 1-2-3 Conjecture and the many
variants that have been proposed in its short but active history.Comment: 30 pages, 2 tables, submitted for publicatio
Distant sum distinguishing index of graphs with bounded minimum degree
For any graph with maximum degree and without isolated
edges, and a positive integer , by we denote the
-distant sum distinguishing index of . This is the least integer for
which a proper edge colouring exists such that
for every pair of distinct vertices
at distance at most in . It was conjectured that
for every . Thus far it
has been in particular proved that if
. Combining probabilistic and constructive approach, we show that this
can be improved to if the
minimum degree of equals at least .Comment: 12 page
Distant total sum distinguishing index of graphs
Let be a proper total colouring of a graph
with maximum degree . We say vertices are sum
distinguished if . By
we denote the least integer admitting such a
colouring for which every , , at distance at most
from each other are sum distinguished in . For every positive integer an
infinite family of examples is known with
. In this paper we prove that
for every integer and
each graph , while .Comment: 10 pages. arXiv admin note: text overlap with arXiv:1703.0037
A note on asymptotically optimal neighbour sum distinguishing colourings
The least admitting a proper edge colouring of a
graph without isolated edges such that for every is denoted by . It
has been conjectured that for every
connected graph of order at least three different from the cycle , where
is the maximum degree of . It is known that for a graph without
isolated edges. We improve this upper bound to using a simpler approach involving a combinatorial
algorithm enhanced by the probabilistic method. The same upper bound is
provided for the total version of this problem as well.Comment: 9 page
Asymptotically optimal neighbour sum distinguishing total colourings of graphs
Consider a simple graph of maximum degree and its proper
total colouring with the elements of the set . The
colouring is said to be \emph{neighbour sum distinguishing} if for every
pair of adjacent vertices , , we have . The least integer for which it exists is denoted
by , hence . On the other hand,
it has been daringly conjectured that just one more label than presumed in the
famous Total Colouring Conjecture suffices to construct such total colouring
, i.e., that for all graphs. We support this
inequality by proving its asymptotic version, . The major part of the construction confirming this relays on a
random assignment of colours, where the choice for every edge is biased by so
called attractors, randomly assigned to the vertices, and the probabilistic
result of Molloy and Reed on the Total Colouring Conjecture itself.Comment: 19 page
Asymptotically optimal bound on the adjacent vertex distinguishing edge choice number
An adjacent vertex distinguishing edge colouring of a graph without
isolated edges is its proper edge colouring such that no pair of adjacent
vertices meets the same set of colours in . We show that such colouring can
be chosen from any set of lists associated to the edges of as long as the
size of every list is at least ,
where is the maximum degree of and is a constant. The proof is
probabilistic. The same is true in the environment of total colourings.Comment: 12 page
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