{"id":195910,"date":"2025-03-01T09:26:52","date_gmt":"2025-03-01T09:26:52","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=195910"},"modified":"2025-03-01T09:26:55","modified_gmt":"2025-03-01T09:26:55","slug":"3is-the-graph-k34-planar","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/01\/3is-the-graph-k34-planar\/","title":{"rendered":"3,Is the graph K3,4 planar"},"content":{"rendered":"\n<p>3,Is the graph K3,4 planar? If yes, give a planar drawing by hand. If not, give a proof of this.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>The graph ( K_{3,4} ) is <strong>not planar<\/strong>. To understand why, we need to use <strong>Kuratowski&#8217;s Theorem<\/strong>, which states that a graph is planar if and only if it does not contain a subgraph that is a <strong>K5<\/strong> (complete graph on 5 vertices) or <strong>K3,3<\/strong> (complete bipartite graph on 3 vertices and 3 vertices).<\/p>\n\n\n\n<p>In the case of ( K_{3,4} ), the graph is a <strong>complete bipartite graph<\/strong> with 3 vertices on one side and 4 vertices on the other side. The number of edges in ( K_{3,4} ) is 12, as each vertex on the left side is connected to every vertex on the right side.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Proof that ( K_{3,4} ) is non-planar:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Kuratowski\u2019s Theorem<\/strong>: To prove ( K_{3,4} ) is non-planar, we need to show that it contains a subgraph that is not planar, specifically ( K_{3,3} ), because a subgraph of ( K_{3,4} ) can be found that is isomorphic to ( K_{3,3} ). We can isolate a set of 3 vertices on the left side of ( K_{3,4} ) and 3 vertices on the right side, and these vertices and their edges form a ( K_{3,3} ), which is non-planar.<\/li>\n\n\n\n<li><strong>( K_{3,3} ) is Non-Planar<\/strong>: It is known that ( K_{3,3} ) is non-planar because it cannot be drawn on a plane without edges crossing.<\/li>\n<\/ol>\n\n\n\n<p>Since ( K_{3,4} ) contains a ( K_{3,3} ) subgraph, by Kuratowski\u2019s Theorem, ( K_{3,4} ) is <strong>non-planar<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Visualization:<\/h3>\n\n\n\n<p>Since ( K_{3,4} ) is non-planar, we can&#8217;t give a planar drawing of it. However, I can show a representation of the graph, which illustrates its structure while respecting the fact that it is non-planar.<\/p>\n\n\n\n<p>Let me generate an image to represent the graph structure of ( K_{3,4} ).<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/03\/image-29.png\" alt=\"\" class=\"wp-image-195911\"\/><\/figure>\n\n\n\n<p>Here is the representation of the graph ( K_{3,4} ), showing the two sets of vertices with edges connecting every vertex from the first set to all vertices in the second set. Since ( K_{3,4} ) contains a subgraph that is isomorphic to ( K_{3,3} ), which is non-planar, the graph as a whole is also non-planar.<\/p>\n\n\n\n<p>This visualization highlights the structure, and it shows why such a graph can&#8217;t be drawn on a plane without edge crossings, confirming that ( K_{3,4} ) is not planar.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>3,Is the graph K3,4 planar? If yes, give a planar drawing by hand. If not, give a proof of this. The Correct Answer and Explanation is : The graph ( K_{3,4} ) is not planar. To understand why, we need to use Kuratowski&#8217;s Theorem, which states that a graph is planar if and only if [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-195910","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195910","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=195910"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195910\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195910"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195910"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195910"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}