73 research outputs found
Distant set distinguishing edge colourings of graphs
We consider the following extension of the concept of adjacent strong edge
colourings of graphs without isolated edges. Two distinct vertices which are at
distant at most in a graph are called -adjacent. The least number of
colours in a proper edge colouring of a graph such that the sets of colours
met by any -adjacent vertices in are distinct is called the -adjacent
strong chromatic index of and denoted by . It has been
conjectured that if is connected of maximum
degree and non-isomorphic to , while Hatami proved that there is
a constant , , such that if
[J. Combin. Theory Ser. B 95 (2005) 246--256]. We conjecture
that a similar statement should hold for any , i.e., that for each positive
integer there exist constants and such that for every graph without an isolated edge and with minimum degree
, and argue that a lower bound on is unavoidable
in such a case (for ). Using the probabilistic method we prove such upper
bound to hold for graphs with , for every and
any fixed , i.e., in particular for regular graphs. We
also support the conjecture by proving an upper bound for graphs with .Comment: 17 page
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 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
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
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
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
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
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
The 1-2-3 Conjecture almost holds for regular graphs
The well-known 1-2-3 Conjecture asserts that the edges of every graph without
isolated edges can be weighted with , and so that adjacent vertices
receive distinct weighted degrees. This is open in general, while it is known
to be possible from the weight set . We show that for regular
graphs it is sufficient to use weights , , , . Moreover, we prove
the conjecture to hold for every -regular graph with .Comment: 15 page
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