Sequence variations of the 123 Conjecture and irregularity strength
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 November 2012
 DOI:
 10.48550/arXiv.1211.0463
 arXiv:
 arXiv:1211.0463
 Bibcode:
 2012arXiv1211.0463S
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 05C15 (Primary) 05C78 (Secondary)
 EPrint:
 Accepted to Discrete Mathematics and Theoretical Computer Science