{"id":97,"date":"2020-04-16T03:06:16","date_gmt":"2020-04-16T03:06:16","guid":{"rendered":"http:\/\/blogs.rollins.edu\/graphembeddings\/?page_id=97"},"modified":"2020-05-05T09:43:12","modified_gmt":"2020-05-05T09:43:12","slug":"embedding-complete-bipartite-graphs-with-cayley-maps","status":"publish","type":"page","link":"https:\/\/blogs.rollins.edu\/graphembeddings\/","title":{"rendered":"Embedding Complete Bipartite Graphs with Cayley Maps"},"content":{"rendered":"\n\n\n\n\n

Our ability to replicate the optimal genera when embedding complete bipartite graphs onto orientable surfaces varied (See Table Below)<\/em>. Though with \"K_{2,2}\" and \"K_{3,3}\" we were able to obtain optimal genera 0 and 1, respectively, we found that it was impossible to obtain an optimal graph embedding for \"K_{5,5}\" and \"K_{7,7}\" using Cayley Maps. The smallest orientable surface \"K_{5,5}\" can embed using this method is a six-holed torus, and \"K_{7,7}\" can embed, at best, an
eight-holed torus. This is likely due to \"K_{5,5}\" and \"K_{7,7}\" being larger, and therefore more complex, than \"K_{2,2}\" and \"K_{3,3}\". As shown in the table below we have yet to determine the minimum genus obtainable using a Cayley Map for \"K_{11,11}\". Because \"K_{11,11}\" is much larger than the other complete bipartite graphs discussed, there are more possible face size combinations to rule out, and, as demonstrated in Theorem 7, doing so can be a lengthy process. <\/p>\n\n\n\n

Though using Cayley Maps as a way to embed complete bipartite graphs may not produce the best genera, \"K_{5,5}\"‘s Cayley Map genus was only three higher than optimal, and \"K_{7,7}\"‘s was only one higher. Therefore, this method can still be useful if simplicity is a greater priority than genus optimality.<\/p>\n\n\n\n

\"\"
Table Comparing Genera Obtained using Cayley Maps to the Optimal Genera.<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"

Our ability to replicate the optimal genera when embedding complete bipartite graphs onto orientable surfaces varied (See Table Below). Though with $K_{2,2}$ and $K_{3,3}$ we were able to obtain optimal genera 0 and 1, respectively, we found that it was impossible to obtain an optimal graph embedding for $K_{5,5}$ and $K_{7,7}$ using Cayley Maps. The […]<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":0,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-97","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/97","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/comments?post=97"}],"version-history":[{"count":36,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/97\/revisions"}],"predecessor-version":[{"id":416,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/97\/revisions\/416"}],"wp:attachment":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/media?parent=97"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}