Edge Induced Weightings of Uniform Hypergraphs and Related Problems
Laars Helenius · ScholarWorks - WMU (Western Michigan University) · 2018
The starting point of the research is the so called 1-2-3 Conjecture formulated in 2004 by Karoński, Luczak, and Thomason. Roughly speaking it says that the edges of any graph can be weighted from {1, 2, 3} so that the induced vertex coloring (as the sum of weights adjacent to a given vertex) is proper. The conjecture has attracted a lot of interest from researchers over the last decade but is still unanswered. More recently, the conjecture has been studied for hypergraphs. The main result of this dissertation shows in particular that an analogous conjecture holds for almost all uniform hypergraphs. Additionally, it also studies how other sets with binary operations, e.g. nite abelian groups, can be used to color hypergraph edges, how hypergraphs are connected by a certain type of path structure, and calculates the threshold probability in a random hypergraph for the appearance of the related cycle structure.