Liangxia Wan
The problem of determining the genus for a graph can be traced back to the Map Color Conjecture proposed by Heawood in 1890, which was later expanded upon by Hilbert and Cohn-Vossen. This conjecture was ultimately established through the contributions of Ringel, Youngs, and several other mathematicians. The determination of genus embeddings is closely linked with other graph invariants, notably influencing the computational complexity of known NP-hard problems. This survey outlines the primary construction techniques and criteria relevant to the genus of a graph. It includes a catalog of most graphs for which the genus is known and introduces a new theorem asserting the validity of the joint trees method for graphs. Additionally, a formal set is proposed, accompanied by relevant results. Interestingly, it is observed that a cubic graph of Hamilton cycle, despite being asymmetric, relates to a formal set of associative surfaces corresponding to all its joint trees, which exhibits symmetry.
@article{20211d2e-f213-4655-be6f-9d216dadf0af,
title={The Genus of a Graph: A Survey},
author={Liangxia Wan},
year={2023},
language={English}
}TY - JOUR TI - The Genus of a Graph: A Survey AU - Liangxia Wan PY - 2023 LA - English ER -
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