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

Colorings, determinants and Alexander polynomials for spatial graphs

A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and $p$-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which $p$ the graph is $p$-colorable, and that a $p$-coloring of a graph corresponds to a representation of the fundamental group of its complement into a metacyclic group $\Gamma(p,m,k)$. We finish by proving some properties of the Alexander polynomial.Comment: 14 pages, 7 figures; version 3 reorganizes the paper, shortens some of the proofs, and improves the results related to representations in metacyclic groups. This is the final version, accepted by Journal of Knot Theory and its Ramification

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