A Decomposition of a Complete Graph with a Hole

• 作者: Roxanne Back Alejandra Brewer Castano Rachel Galindo Jessica Finocchiaro
• 来源期刊: 离散数学期刊(英文)
• 学科: 数学
• 关键词: Graph Decomposition Combinatorial Design Complete Graph with a Hole
• 年，卷（期）: 2021,（1）
• 页码范围: 1-12
• 页数: 12页
• 分类号: O15

摘要

<div style="text-align:justify;"> <span style="font-family:Verdana;">In the field of design theory, the most well-known design is a Steiner Triple System. In general, a G-design on H is an edge-disjoint decomposition of H into isomorphic copies of G. In a Steiner Triple system, a complete graph is decomposed into triangles. In this paper we let H be a complete graph with a hole and G be a complete graph on four vertices minus one edge, also referred to as a <img alt="" src="Edit_e69ee166-4bbc-48f5-8ba1-b446e7d3738c.png" /> . A complete graph with a hole, <img alt="" src="Edit_558c249b-55e8-4f3b-a043-e36d001c4250.png" />, consists of a complete graph on <em>d</em> vertices, <img alt="" src="Edit_cb1772f7-837c-4aea-b4a6-cb38565f5a8b.png" />, and a set of independent vertices of size<em> v, V,</em> where each vertex in <em>V</em> is adjacent to each vertex in <img alt="" src="Edit_cb1772f7-837c-4aea-b4a6-cb38565f5a8b.png" />. When <em>d</em> is even, we give two constructions for the decomposition of a complete graph with a hole into copies of <img alt="" src="Edit_e69ee166-4bbc-48f5-8ba1-b446e7d3738c.png" /> : the Alpha-Delta Construction, and the Alpha-Beta-Delta Construction. By restricting <em>d</em> and <em>v</em> so that <img alt="" src="Edit_6bb9e3b4-1769-4b28-bf89-bc97c47c637e.png" /><span style="white-space:nowrap;"> </span>, we are able to resolve both of these cases for a subset of <img alt="" src="Edit_558c249b-55e8-4f3b-a043-e36d001c4250.png" />using difference methods and 1-factors.</span> </div>