An edge colouring of multigraphs

We consider a strict k-colouring of a multigraph G as a surjection f from the vertex set of G into a set of colours {1,2,…,k} such that, for every non-pendant vertex χ of G, there exist at least two edges incident to χ and coloured by the same colour. The maximum number of colours in a strict edge colouring of G is called the upper chromatic index of G and is denoted by χ(G). In this paper we prove some results about it

Rainbow matchings in properly-coloured multigraphs

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by $n$ colours with at least $n + 1$ edges of each colour there must be a matching that uses each colour exactly once. In this paper we consider the same question without the bipartiteness assumption. We show that in any multigraph with edge multiplicities $o(n)$ that is properly edge-coloured by $n$ colours with at least $n + o(n)$ edges of each colour there must be a matching of size $n-O(1)$ that uses each colour at most once.Comment: 7 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