Extremal Problems for Partitions of Edge Sets of Graphs

Abstract

This thesis considers three families of problems in graph theory about partitions of the edge sets of graphs (also known as graph decompositions). The first family we address consists of induced Ramsey number problems. Induced Ramsey numbers generalize ordinary Ramsey numbers. The induced Ramsey number of a pair of graphs G and H is the smallest n such that there exists a graph F on n vertices such that any partition of the edge set of F into two sets yields an induced copy of G in one set of the partition or H in the other set. We prove several results on the induced Ramsey number of P3 with other graphs. We also show that the induced Ramsey number can differ asymptotically from a related parameter, the weak induced Ramsey number. Then, we look at a family of problems where we prove results on the existence of a partition of the edge set of a planar graph into a forest and another graph, where the maximum degree of the other graph is not too high. In particular, we show that a planar graph of girth 9 or higher decomposes into a forest and a matching. We also show that a planar graph that has no cycles of length 4 decomposes into a forest and a graph with maximum degree at most 5. Finally, we prove that the number of perfect matchings in a certain family of planar graphs is divisible by 3. A perfect matching can be regarded as a partition of the edge set into two sets such that in the first set each vertex has degree exactly 1.