{"id":14566,"date":"2016-03-01T03:06:41","date_gmt":"2016-03-01T03:06:41","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=14566"},"modified":"2016-03-01T03:06:41","modified_gmt":"2016-03-01T03:06:41","slug":"math-comp-sci-graph-theory-hamilton-cycle","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/math-comp-sci-graph-theory-hamilton-cycle\/","title":{"rendered":"Math & Comp Sci: Graph theory: Hamilton cycle"},"content":{"rendered":"

The tutor proves the existence of a Hamilton cycle under certain conditions.<\/h1>\n

A Hamilton cycle, named for the Irish mathematician Sir William Rowan Hamilton, is a path that visits each vertex of a graph exactly once before returning to the start point. On page 586 of Grimaldi, the reader is asked to prove the following:<\/p>\n

A graph with n ≥ 3 vertices, with deg ≥ n\/2 for each one, has a Hamilton cycle.<\/strong><\/p>\n

Proof:<\/p>\n

We will prove by induction. For the case of 3 vertices, each with degree 2, there is obviously a Hamilton cycle:<\/p>\n

<\/p>\n

Let’s now assume a graph of n vertices with a Hamilton cycle. If we add another vertex, |V| = n + 1. The the new vertex itself has deg ≥ (n+1)\/2, which means it joins to more than half of the n orginal points. Perhaps only because they form a cycle, two of those points must be adjacent. Let’s imagine those points are Pq<\/sub> and Pr<\/sub>. Then the new point can be included in the cycle between Pq<\/sub> and Pr<\/sub>. So, the new graph with n + 1 vertices also has a Hamilton cycle.<\/p>\n

I’ll be discussing more graph theory in future posts:)<\/p>\n

Source:<\/p>\n

Grimaldi, Ralph P. Discrete and Combinatorial Mathematics<\/u>. Don Mills: Addison-
  Wesley, 1994.<\/p>\n

Jack of Oracle Tutoring by Jack and Diane,<\/a> Campbell River, BC.<\/p>\n","protected":false},"excerpt":{"rendered":"

The tutor proves the existence of a Hamilton cycle under certain conditions. A Hamilton cycle, named for the Irish mathematician Sir William Rowan Hamilton, is a path that visits each vertex of a graph exactly once before returning to the …<\/p>\n

Math & Comp Sci: Graph theory: Hamilton cycle<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[105,3],"tags":[1444,1465,1466],"class_list":["post-14566","post","type-post","status-publish","format-standard","hentry","category-computer-science","category-math","tag-graph-theory","tag-hamilton-cycle","tag-proof-of-existence-of-hamilton-cycle"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/14566","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/comments?post=14566"}],"version-history":[{"count":8,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/14566\/revisions"}],"predecessor-version":[{"id":14574,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/14566\/revisions\/14574"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=14566"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=14566"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=14566"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}