Saturation Problems on Graphs
| dc.contributor.author | Ogden, Shannon | |
| dc.contributor.supervisor | Morrison, Natasha | |
| dc.contributor.supervisor | Mynhardt, Kieka | |
| dc.date.accessioned | 2023-08-25T22:55:23Z | |
| dc.date.available | 2023-08-25T22:55:23Z | |
| dc.date.copyright | 2023 | en_US |
| dc.date.issued | 2023-08-25 | |
| dc.degree.department | Department of Mathematics and Statistics | en_US |
| dc.degree.level | Master of Science M.Sc. | en_US |
| dc.description.abstract | In this thesis, we consider two variations on classical saturation problems in extremal graph theory: rainbow saturation and weak saturation. An edge-coloured graph G is rainbow if every edge in G receives a distinct colour. Given a graph H, an edge-coloured graph G is H-rainbow-saturated if G does not contain a rainbow copy of H, but the addition of any non-edge to G, in any colour, creates a rainbow copy of H. The rainbow saturation number of H, denoted by rsat(n,H), is the minimum number of edges in an H-rainbow saturated graph on n vertices. In Chapter 2, we prove that, like ordinary saturation numbers, the rainbow saturation number of every graph H is linear in n. This result confirms a conjecture of Girao, Lewis, and Popielarz. In Chapter 3, we consider a specific type of weak saturation known as r-bond bootstrap percolation. In the r-bond bootstrap percolation process on a graph G, we start with a set of initially infected edges of G, and consider all other edges in G to be healthy. At each subsequent step in the process, the infection spreads to a healthy edge if at least one of its endpoints is incident with at least r infected edges. Once an edge is infected, it remains infected indefinitely. If a set of initially infected edges will eventually infect all of E(G), we refer to it as an r-percolating set of G. Define m_e(G,r) to be the minimum number of edges in an r-percolating set of G. Recently, Hambardzumyan, Hatami, and Qian introduced a clever new polynomial method, which they used to provide recursive formulas for m_e(G,r) when G is either a d-dimensional torus or a d-dimensional grid. We push this polynomial method further, in order to determine m_e(G,r) for certain other graphs G. In particular, we provide recursive formulas for m_e(G,r) when G is a Cartesian product of stars or a Cartesian product of joined cycles (cycles with a single chord). We also give upper and lower bounds on m_e(G,r) when G is a Cartesian product of a tree with any graph H, and examine the conditions under which these bounds match. | en_US |
| dc.description.scholarlevel | Graduate | en_US |
| dc.identifier.uri | http://hdl.handle.net/1828/15290 | |
| dc.language | English | eng |
| dc.language.iso | en | en_US |
| dc.rights | Available to the World Wide Web | en_US |
| dc.subject | Graph Theory | en_US |
| dc.subject | Saturation | en_US |
| dc.subject | Rainbow Saturation | en_US |
| dc.subject | Weak Saturation | en_US |
| dc.subject | Bootstrap Percolation | en_US |
| dc.title | Saturation Problems on Graphs | en_US |
| dc.type | Thesis | en_US |