Index 3 biembeddings of the complete graphs
Juvenal F. Barajas, Timothy Sun
TL;DR
This paper presents a novel method for decomposing complete graphs into pairs of edge-disjoint triangulations of orientable surfaces using index 3 current graphs, solving a generalized Earth-Moon problem.
Contribution
It introduces a new construction technique for biembeddings of complete graphs into orientable surfaces based on index 3 current graphs, expanding previous methods.
Findings
Decompositions exist for complete graphs with 24s+21 vertices.
Pairs of embeddings are derived from index 3 current graphs.
Solves a generalized Earth-Moon problem for certain surfaces.
Abstract
We show that the complete graphs on vertices have decompositions into two edge-disjoint subgraphs, each of which triangulates an orientable surface. The special case where the two surfaces are homeomorphic solves a generalized Earth-Moon problem for that surface. Unlike previous constructions, these pairs of triangular embeddings are derived from index 3 current graphs.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · graph theory and CDMA systems