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Embedding and Coloring Designs.

Baumann, Stacie.

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Embedding and Coloring Designs.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Embedding and Coloring Designs./
作者:	Baumann, Stacie.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2023,
面頁冊數:	65 p.
附註:	Source: Dissertations Abstracts International, Volume: 85-03, Section: A.
Contained By:	Dissertations Abstracts International85-03A.
標題:	Graphs. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30561853
ISBN:	9798380261944

Embedding and Coloring Designs.
Baumann, Stacie.

Embedding and Coloring Designs. - Ann Arbor : ProQuest Dissertations & Theses, 2023 - 65 p.

Source: Dissertations Abstracts International, Volume: 85-03, Section: A.

Thesis (Ph.D.)--Auburn University, 2023.

This item must not be sold to any third party vendors.

This dissertation focuses on two problems in design theory. Techniques from graph theory are frequently utilized and therefore the proofs may also be of interest to graph theorists.The first problem focuses on completing partial latin squares with prescribed diagonals. Necessary and sufficient numerical conditions are known for the embedding of an incomplete latin square L of order n into a latin square T of order t ≥ 2n + 1 in which each symbol is prescribed to occur in a given number of cells on the diagonal of T outside of L. This includes the classic case where T is required to be idempotent. If t < 2n then no such numerical sufficient conditions exist since it is known that the arrangement of symbols within the given incomplete latin square can determine the embeddability. All known examples where the arrangement is a factor share the common feature that one symbol is prescribed to appear exactly once in the diagonal of T outside of L. We show if the prescribed diagonal contains a symbol required to appear exactly once on the diagonal of T outside of L and t ≤ 2n, then there always exists a incomplete latin square satisfying the known numerical necessary conditions that is non-embeddable. Also, we solve a conjecture made over 30 years ago stating it is only this feature that prevents numerical conditions sufficing for all t ≥ n. Thus providing necessary and sufficient numerical conditions for the embedding of an incomplete latin square L of order n into a latin square T of order t for all t ≥ n in which the diagonal of T outside of L is prescribed in the case where no symbol is required to appear exactly once in the diagonal of T outside of L.The second problem focuses on (not necessarily proper) s-edge-colorings of Kv in which, for all u ∈ V (Kv), the edges incident with u are colored using exactly p colors. In the spirit of proper edge-colorings, such (s, p)-edge-colorings are required to be equitable: the edges at each vertex are shared evenly among p colors. First, results related to the existence of equitable (s, p)-edge-colorings of Kv and future directions related to equitable (s, p)-edgecolorings of λKv are discussed. Then, the structure of equitable (s, p)-edge-colorings of Kv is addressed, particularly, the number of vertices at which each color appears. Results are obtained determining how large and how small these numbers can be. Results concerning equitable (s, p)-C4-colorings of Kv − Ffollow as corollaries.

ISBN: 9798380261944Subjects--Topical Terms:

3685259
Graphs.

Embedding and Coloring Designs.
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500 $a Source: Dissertations Abstracts International, Volume: 85-03, Section: A.
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520 $a This dissertation focuses on two problems in design theory. Techniques from graph theory are frequently utilized and therefore the proofs may also be of interest to graph theorists.The first problem focuses on completing partial latin squares with prescribed diagonals. Necessary and sufficient numerical conditions are known for the embedding of an incomplete latin square L of order n into a latin square T of order t ≥ 2n + 1 in which each symbol is prescribed to occur in a given number of cells on the diagonal of T outside of L. This includes the classic case where T is required to be idempotent. If t < 2n then no such numerical sufficient conditions exist since it is known that the arrangement of symbols within the given incomplete latin square can determine the embeddability. All known examples where the arrangement is a factor share the common feature that one symbol is prescribed to appear exactly once in the diagonal of T outside of L. We show if the prescribed diagonal contains a symbol required to appear exactly once on the diagonal of T outside of L and t ≤ 2n, then there always exists a incomplete latin square satisfying the known numerical necessary conditions that is non-embeddable. Also, we solve a conjecture made over 30 years ago stating it is only this feature that prevents numerical conditions sufficing for all t ≥ n. Thus providing necessary and sufficient numerical conditions for the embedding of an incomplete latin square L of order n into a latin square T of order t for all t ≥ n in which the diagonal of T outside of L is prescribed in the case where no symbol is required to appear exactly once in the diagonal of T outside of L.The second problem focuses on (not necessarily proper) s-edge-colorings of Kv in which, for all u ∈ V (Kv), the edges incident with u are colored using exactly p colors. In the spirit of proper edge-colorings, such (s, p)-edge-colorings are required to be equitable: the edges at each vertex are shared evenly among p colors. First, results related to the existence of equitable (s, p)-edge-colorings of Kv and future directions related to equitable (s, p)-edgecolorings of λKv are discussed. Then, the structure of equitable (s, p)-edge-colorings of Kv is addressed, particularly, the number of vertices at which each color appears. Results are obtained determining how large and how small these numbers can be. Results concerning equitable (s, p)-C4-colorings of Kv − Ffollow as corollaries.
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