Graph Decompositions

Abstract

The topic of this PhD thesis is graph decompositions. While there exist various kinds of decompositions, this thesis focuses on three problems concerning edgedecompositions. Given a family of graphs H we ask the following question: When can the edge-set of a graph be partitioned so that each part induces a subgraph isomorphic to a member of H? Such a decomposition is called an H-decomposition. Apart from the existence of an H-decomposition, we are also interested in the number of parts needed in an H-decomposition. Firstly, we show that for every tree T there exists a constant k(T) such that every k(T)-edge-connected graph whose size is divisible by the size of T admits a T-decomposition. This proves a conjecture by Barát and Thomassen from 2006.Moreover, we introduce a new arboricity notion where we restrict the diameter of the trees in a decomposition into forests. We conjecture that for every natural number k there exists a natural number d(k) such that the following holds: If G can be decomposed into k forests, then G can be decomposed into k + 1 forests in which each tree has diameter at most d(k). We verify this conjecture fork ≤ 3. As an application we show that every 6-edge-connected planar graph contains two edge-disjoint 18/19 -thin spanning trees.Finally, we make progress on a conjecture by Baudon, Bensmail, Przybyło, and Wozniak stating that if a graph can be decomposed into locally irregular graphs, then there exists such a decomposition with at most 3 parts. We show that this conjecture is true if the number 3 is replaced by 328, establishing the first constant upper bound for this problem