Mathematics, Department of


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A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Stephen G. Hartke. Lincoln, Nebraska: May, 2014

Copyright (c) 2014 James Carraher


This thesis focuses on determining when a graph with additional structure contains certain subgraphs, particularly circuits, cycles, or trees. The specific problems and presented results include a blend of many fundamental graph theory concepts such as edge-coloring, routing problems, decomposition problems, and containing cycles of various lengths. The three primary chapters in this thesis address the problems of finding eulerian circuits with additional restrictions, decomposing the edge-colored complete graph K_n into rainbow spanning trees, and showing a 4-connected claw-free and N(3,2,1)-free graph is pancyclic.

Adviser: Stephen G. Hartke