23 research outputs found
The inapproximability for the (0,1)-additive number
An
{\it additive labeling} of a graph is a function , such that for every two adjacent vertices and of , ( means that is joined to ). The {\it additive number} of ,
denoted by , is the minimum number such that has a additive
labeling . The {\it additive
choosability} of a graph , denoted by , is the smallest
number such that has an additive labeling for any assignment of lists
of size to the vertices of , such that the label of each vertex belongs
to its own list.
Seamone (2012) \cite{a80} conjectured that for every graph , . We give a negative answer to this conjecture and we show that
for every there is a graph such that .
A {\it -additive labeling} of a graph is a function , such that for every two adjacent vertices and
of , .
A graph may lack any -additive labeling. We show that it is -complete to decide whether a -additive labeling exists for
some families of graphs such as perfect graphs and planar triangle-free graphs.
For a graph with some -additive labelings, the -additive
number of is defined as where is the set of -additive labelings of .
We prove that given a planar graph that admits a -additive labeling, for
all , approximating the -additive number within is -hard.Comment: 14 pages, 3 figures, Discrete Mathematics & Theoretical Computer
Scienc
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