On the Depression of Graphs
| dc.contributor.author | Schurch, Mark | |
| dc.contributor.supervisor | Mynhardt, C. M. | |
| dc.date.accessioned | 2013-04-17T22:17:38Z | |
| dc.date.available | 2013-04-17T22:17:38Z | |
| dc.date.copyright | 2013 | en_US |
| dc.date.issued | 2013-04-17 | |
| dc.degree.department | Department of Mathematics and Statistics | |
| dc.degree.level | Doctor of Philosophy Ph.D. | en_US |
| dc.description.abstract | An edge ordering of a graph G = (V,E) is an injection f : E → R, where R denotes the set of real numbers. A path in G for which the edge ordering f increases along its edge sequence is called an f-ascent; an f-ascent is maximal if it is not contained in a longer f-ascent. The depression of G is the smallest integer k such that any edge ordering f has a maximal f-ascent of length at most k. In this dissertation we discuss various results relating to the depression of a graph. We determine a formula for the depression of the class of trees known as double spiders. A k-kernel of a graph G is a set of vertices U ⊆ V (G) such that for any edge ordering f of G there exists a maximal f-ascent of length at most k which neither starts nor ends in U. We study the concept of k-kernels and discuss related depression results, including an improved upper bound for the depression of trees. We include a characterization of the class of graphs with depression three and without adjacent vertices of degree three or higher, and also construct a large class of graphs with depression three which contains graphs with adjacent vertices of high degree. Lastly, we apply the concept of ascents to edge colourings using possibly fewer than |E(G)| colours (integers). We consider the problem of determining the minimum number of colours for which there exists an edge colouring such that the length of a shortest maximal path of edges with increasing colors has a given length. | en_US |
| dc.description.proquestcode | 0405 | en_US |
| dc.description.scholarlevel | Graduate | en_US |
| dc.identifier.uri | http://hdl.handle.net/1828/4527 | |
| dc.language | English | eng |
| dc.language.iso | en | en_US |
| dc.rights.temp | Available to the World Wide Web | en_US |
| dc.subject | graph theory | en_US |
| dc.subject | edge labellings | en_US |
| dc.subject | edge colourings | en_US |
| dc.title | On the Depression of Graphs | en_US |
| dc.type | Thesis | en_US |