3,663 research outputs found
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
Linear bounds on nowhere-zero group irregularity strength and nowhere-zero group sum chromatic number of graphs
We investigate the \textit{group irregularity strength}, , of a
graph, i.e. the least integer such that taking any Abelian group
of order , there exists a function so that the sums of edge labels incident with every vertex are
distinct. So far the best upper bound on for a general graph was
exponential in , where is the order of and denotes the number
of its components. In this note we prove that is linear in , namely
not greater than . In fact, we prove a stronger result, as we additionally
forbid the identity element of a group to be an edge label or the sum of labels
around a vertex.
We consider also locally irregular labelings where we require only sums of
adjacent vertices to be distinct. For the corresponding graph invariant we
prove the general upper bound: (where
is the coloring number of ) in the case when we do not use the identity
element as an edge label, and a slightly worse one if we additionally forbid it
as the sum of labels around a vertex. In the both cases we also provide a sharp
upper bound for trees and a constant upper bound for the family of planar
graphs
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
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 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
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
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
On the neighbour sum distinguishing index of planar graphs
Let be a proper edge colouring of a graph with integers
. Then , while by Vizing's theorem, no more than
is necessary for constructing such . On the course of
investigating irregularities in graphs, it has been moreover conjectured that
only slightly larger , i.e., enables enforcing additional
strong feature of , namely that it attributes distinct sums of incident
colours to adjacent vertices in if only this graph has no isolated edges
and is not isomorphic to . We prove the conjecture is valid for planar
graphs of sufficiently large maximum degree. In fact even stronger statement
holds, as the necessary number of colours stemming from the result of Vizing is
proved to be sufficient for this family of graphs. Specifically, our main
result states that every planar graph of maximum degree at least which
contains no isolated edges admits a proper edge colouring
such that for every edge of .Comment: 22 page
Group twin coloring of graphs
For a given graph , the least integer such that for every
Abelian group of order there exists a proper edge labeling
so that for each edge is called the \textit{group twin
chromatic index} of and denoted by . This graph invariant is
related to a few well-known problems in the field of neighbor distinguishing
graph colorings. We conjecture that for all graphs
without isolated edges, where is the maximum degree of , and
provide an infinite family of connected graph (trees) for which the equality
holds. We prove that this conjecture is valid for all trees, and then apply
this result as the base case for proving a general upper bound for all graphs
without isolated edges: , where
denotes the coloring number of . This improves the best known
upper bound known previously only for the case of cyclic groups
New results on eigenvalues and degree deviation
Let be a graph. In a famous paper Collatz and Sinogowitz had proposed to
measure its deviation from regularity by the difference of the (adjacency)
spectral radius and the average degree: .
We obtain here a new upper bound on which seems to consistently
outperform the best known upper bound to date, due to Nikiforov. The method of
proof may also be of independent interest, as we use notions from numerical
analysis to re-cast the estimation of as a special case of the
estimation of the difference between Rayleigh quotients of proximal vectors
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