Total weight choosability in Hypergraphs

A total weighting of the vertices and edges of a hypergraph is called vertex-coloring if the total weights of the vertices yield a proper coloring of the graph, i.e., every edge contains at least two vertices with different weighted degrees. In this note we show that such a weighting is possible if every vertex has two, and every edge has three weights to choose from, extending a recent result on graphs to hypergraphs

The 1-2-3 Conjecture for Hypergraphs

A weighting of the edges of a hypergraph is called vertex-coloring if the weighted degrees of the vertices yield a proper coloring of the graph, i.e., every edge contains at least two vertices with different weighted degrees. In this paper we show that such a weighting is possible from the weight set {1,2,...,r+1} for all hypergraphs with maximum edge size r>3 and not containing edges solely consisting of identical vertices. The number r+1 is best possible for this statement. Further, the weight set {1,2,3,4,5} is sufficient for all hypergraphs with maximum edge size 3, up to some trivial exceptions.Comment: 12 page

Few Long Lists for Edge Choosability of Planar Cubic Graphs

It is known that every loopless cubic graph is 4-edge choosable. We prove the following strengthened result. Let G be a planar cubic graph having b cut-edges. There exists a set F of at most 5b/2 edges of G with the following property. For any function L which assigns to each edge of F a set of 4 colours and which assigns to each edge in E(G)-F a set of 3 colours, the graph G has a proper edge colouring where the colour of each edge e belongs to L(e).Comment: 14 pages, 1 figur