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Total Edge Irregularity Strength for Graphs
An edge irregular total -labelling of a graph is a labelling of the vertices and the edges of
in such a way that any two different edges have distinct weights. The
weight of an edge , denoted by , is defined as the sum of the label
of and the labels of two vertices which incident with , i.e. if ,
then . The minimum for which has an edge
irregular total -labelling is called the total edge irregularity strength of
In this paper, we determine total edge irregularity of connected and
disconnected graphs
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
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