29 research outputs found
Total Vertex Irregularity Strength of Forests
We investigate a graph parameter called the total vertex irregularity
strength (), i.e. the minimal such that there is a labeling of the edges and vertices of giving
distinct weighted degrees for every
pair of vertices of . We prove that for
every forest with no vertices of degree 2 and no isolated vertices, where
is the number of pendant vertices in . Stronger results for trees were
recently proved by Nurdin et al.Comment: The stronger results for trees were recently proved by Nurdin et al.
(Nurdin, Baskoro E.T., Salman A.N.M., Gaos N.N., On the Total Vertex
Irregularity Strength of Trees, Discrete Mathematics 310 (2010), 3043-3048.).
However we decided to publish our paper for two reasons. Firstly, we consider
more general case of forests, not only trees. Secondly, we use different
proof techniqu
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
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
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
Product irregularity strength of graphs with small clique cover number
For a graph without isolated vertices and without isolated edges, a
product-irregular labelling , first
defined by Anholcer in 2009, is a labelling of the edges of such that for
any two distinct vertices and of the product of labels of the edges
incident with is different from the product of labels of the edges incident
with . The minimal for which there exist a product irregular labeling is
called the product irregularity strength of and is denoted by .
Clique cover number of a graph is the minimum number of cliques that partition
its vertex-set. In this paper we prove that connected graphs with clique cover
number or have the product-irregularity strength equal to , with
some small exceptions
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
On the neighbour sum distinguishing index of graphs with bounded maximum average degree
A proper edge -colouring of a graph is an assignment of colours to the edges of the graph such that no two
adjacent edges are associated with the same colour. A neighbour sum
distinguishing edge -colouring, or nsd -colouring for short, is a proper
edge -colouring such that for
every edge of . We denote by the neighbour sum
distinguishing index of , which is the least integer such that an nsd
-colouring of exists. By definition at least maximum degree,
colours are needed for this goal. In this paper we prove that for any graph without isolated edges and with , .Comment: 10 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