Mathematics Commons^™

Extremal Results For Peg Solitaire On Graphs, Aaron D. Gray

Electronic Theses and Dissertations

In a 2011 paper by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. An open problem from that paper is to determine the minimum number of edges necessary for a graph with a fixed number of vertices to be solvable. This thesis provides new bounds on this number. It also provides necessary and sufficient conditions for two families of graphs to be solvable, along with criticality results, and the maximum number of pegs that can be left in each of the two graph families.

Packings And Coverings Of Complete Graphs With A Hole With The 4-Cycle With A Pendant Edge, Yan Xia

Electronic Theses and Dissertations

In this thesis, we consider packings and coverings of various complete graphs with the 4-cycle with a pendant edge. We consider both restricted and unrestricted coverings. Necessary and sufficient conditions are given for such structures for (1) complete graphs K_v, (2) complete bipartite graphs K_m,n, and (3) complete graphs with a hole K(v,w).

Peg Solitaire On Trees With Diameter Four, Clayton A. Walvoort

Electronic Theses and Dissertations

In a paper by Beeler and Hoilman, the traditional game of peg solitaire is generalized to graphs in the combinatorial sense. One of the important open problems in this paper was to classify solvable trees. In this thesis, we will give necessary and sufficient conditions for the solvability for all trees with diameter four. We also give the maximum number of pegs that can be left on such a graph under the restriction that we jump whenever possible.

Very Cost Effective Partitions In Graphs, Inna Vasylieva

Electronic Theses and Dissertations

For a graph G=(V,E) and a set of vertices S, a vertex v in S is said to be very cost effective if it is adjacent to more vertices in V -S than in S.

A bipartition pi={S, V- S} is called very cost effective if both S and V- S are very cost effective sets. Not all graphs have a very cost effective bipartition, for example, the complete graphs of odd order do not. We consider several families of graphs G, including Cartesian products and cacti graphs, to determine whether G has a very cost effective bipartition.

Restricted And Unrestricted Coverings Of Complete Bipartite Graphs With Hexagons, Wesley M. Surber

Electronic Theses and Dissertations

A minimal covering of a graph G with isomorphic copies of graph H is a set {H₁, H₂, H₃, ... , H_n} where H_i is isomorphic to H, the vertex set of H_i is a subset of G, the edge set of G is a subset of the union of H_i's, and the cardinality of the union of H_i's minus G is minimum. Some studies have been made of covering the complete graph in which case an added condition of the edge set of H_{i …}