{"id":314,"date":"2020-04-21T09:53:09","date_gmt":"2020-04-21T09:53:09","guid":{"rendered":"http:\/\/blogs.rollins.edu\/graphembeddings\/?page_id=314"},"modified":"2020-04-21T09:53:09","modified_gmt":"2020-04-21T09:53:09","slug":"euler-characteristic-genus","status":"publish","type":"page","link":"https:\/\/blogs.rollins.edu\/graphembeddings\/embedding-complete-bipartite-graphs-with-cayley-maps\/euler-characteristic-genus\/","title":{"rendered":"Euler Characteristic & Genus"},"content":{"rendered":"\n\n\n

Each Cayley Map has an Euler Characteristic, \"\chi\", such that \"\chi=v-e+f\", where \"v\" is the number of vertices, \"e\" is the number of edges, and \"f\" is the number of faces. The \"\chi\" of different Cayley Maps is the same if the Cayley Maps embed the same surface.<\/p>\n\n\n\n

The genus, \"g\", is equal to the number of holes in the surface. Spheres have a genus of 0, one-holed toruses have a genus of 1, two-holed toruses have a genus of 2, etc. The genus of a graph can be found using the formula \"\chi=2-2g\".<\/p>\n\n\n\n

\n\n\n\n

According to Ringel, for any positive integer \"n\", the genus of the complete bipartite graph \"K_{n,n}\" is \"L(n,n)\", where \"L(n,n)\" if the ceiling of \"\frac{(n-2)^2}{4}\". Equations 2 and 3 show the formulas for even and odd integers, specifically.<\/p>\n\n\n\n

Equation 2.<\/strong> For \"n_e\", where \"n_e\" is an even integer, \"L(n_e,n_e)=(\frac{n_e}{2})^2-n_e+1\".<\/p>\n\n\n\n

Equation 3.<\/strong> For \"n_o\", where \"n_o\" is an odd integer, \"L(n_o,n_o)=\frac{n_o^2+3}{4}-n_o+1\".<\/p>\n","protected":false},"excerpt":{"rendered":"

Each Cayley Map has an Euler Characteristic, $\\chi$, such that $\\chi=v-e+f$, where $v$ is the number of vertices, $e$ is the number of edges, and $f$ is the number of faces. The $\\chi$ of different Cayley Maps is the same if the Cayley Maps embed the same surface. The genus, $g$, is equal to the […]<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":97,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-314","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/314","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=314"}],"version-history":[{"count":2,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/314\/revisions"}],"predecessor-version":[{"id":336,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/314\/revisions\/336"}],"up":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/97"}],"wp:attachment":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/media?parent=314"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}