Western Michigan University ScholarWorks at WMU Dissertations Graduate College 1-2011 Hamiltonicity and Connectivity in Distance-Colored Graphs Kyle C. Kolasinski Western Michigan University Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Applied Mathematics Commons, and the Mathematics Commons Recommended Citation Kolasinski, Kyle C., "Hamiltonicity and Connectivity in Distance-Colored Graphs" (2011). Dissertations. 431. https://scholarworks.wmich.edu/dissertations/431 This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact
[email protected]. Hamiltonicity and Connectivity in Distance-Colored Graphs by Kyle C. Kolasinski A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment for the requirement for the Degree of Doctor of Philosophy Department of Mathematics Advisor: Ping Zhang, Ph.D. Western Michigan University Kalamazoo, Michigan June 2011 Hamiltonicity and Connectivity in Distance-Colored Graphs Kyle Kolasinski, Ph.D. Western Michigan University, 2011 For a connected graph G and a positive integer fc, the fcth power Gk of G is the graph with V(Gk) — V(G) where uv E E(Gk) if the distance dc{u^v) between u and v is at most fc. The edge coloring of Gk defined by assigning each edge uv of Gk the color da(u,v) produces an edge-colored graph Gk called a distance-colored graph. A distance-colored graph Gk is Hamiltonian-colored if Gk contains a properly colored Hamiltonian cycle.