Sequence variations of the 1-2-3 Conjecture and irregularity strength

TL;DR

This paper explores variations of the 1-2-3 Conjecture and irregularity strength, focusing on sequence-based vertex colourings with list constraints, providing bounds for different graph classes and ordering conditions.

Contribution

It introduces new bounds on list sizes needed for sequence colourings under different ordering constraints and extends methods to irregularity strength variations.

Findings

Bounded list sizes for sequence colourings with arbitrary edge orderings.

Established list size bounds for graphs with large minimum degree.

Extended techniques to a list variation of 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.