{"id":215745,"date":"2025-05-17T17:23:08","date_gmt":"2025-05-17T17:23:08","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=215745"},"modified":"2025-05-17T17:23:10","modified_gmt":"2025-05-17T17:23:10","slug":"give-an-example-of-a-graph-g-with-the-following-properties-or-explain-why-no-such-example-exists","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/17\/give-an-example-of-a-graph-g-with-the-following-properties-or-explain-why-no-such-example-exists\/","title":{"rendered":"Give an example of a graph G with the following properties or explain why no such example exists."},"content":{"rendered":"\n<p><strong>Give an example of a graph G with the following properties or explain why no such example exists.<\/strong> Here d(G) denotes the minimum degree of a graph G. i, ii, K(G) = 2, ?(G) &#8211; 3 and d(G) = 4 K(G) = 3, ?(G) = 2 and d(G) = 4<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>We are asked whether there exists a graph GG that satisfies the following properties. Let&#8217;s examine each case:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>i.<\/strong> \u03ba(G)=2\\kappa(G) = 2, \u03bb(G)=3\\lambda(G) = 3, and \u03b4(G)=4\\delta(G) = 4<\/h3>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03ba(G)\\kappa(G): vertex-connectivity \u2013 the minimum number of vertices whose removal disconnects the graph.<\/li>\n\n\n\n<li>\u03bb(G)\\lambda(G): edge-connectivity \u2013 the minimum number of edges whose removal disconnects the graph.<\/li>\n\n\n\n<li>\u03b4(G)\\delta(G): minimum degree \u2013 the smallest degree among all vertices in the graph.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 <strong>Answer (i): Yes, such a graph exists.<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Explanation:<\/strong><\/h4>\n\n\n\n<p>By Whitney&#8217;s inequality: \u03ba(G)\u2264\u03bb(G)\u2264\u03b4(G)\\kappa(G) \\leq \\lambda(G) \\leq \\delta(G)<\/p>\n\n\n\n<p>So if \u03b4(G)=4\\delta(G) = 4, then the edge-connectivity \u03bb(G)\\lambda(G) can be less than 4, and similarly, \u03ba(G)\\kappa(G) can be less than \u03bb(G)\\lambda(G). Therefore, having \u03ba(G)=2\\kappa(G) = 2, \u03bb(G)=3\\lambda(G) = 3, and \u03b4(G)=4\\delta(G) = 4 <strong>does not violate<\/strong> Whitney&#8217;s inequality.<\/p>\n\n\n\n<p><strong>Example Construction:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Take two cliques of 5 vertices (each is K5K_5) with all nodes having degree 4.<\/li>\n\n\n\n<li>Connect the cliques with 3 edges between distinct pairs of nodes (one in each clique).<\/li>\n\n\n\n<li>Remove any two vertices \u2014 the graph remains connected (so \u03ba(G)=2\\kappa(G) = 2).<\/li>\n\n\n\n<li>But removing the three connecting edges disconnects the two K5K_5 parts (so \u03bb(G)=3\\lambda(G) = 3).<\/li>\n\n\n\n<li>Each node in K5K_5 has degree 4, satisfying \u03b4(G)=4\\delta(G) = 4.<\/li>\n<\/ul>\n\n\n\n<p>So such a graph <strong>does exist<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>ii.<\/strong> \u03ba(G)=3\\kappa(G) = 3, \u03bb(G)=2\\lambda(G) = 2, and \u03b4(G)=4\\delta(G) = 4<\/h3>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u274c <strong>Answer (ii): No, such a graph does not exist.<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Explanation:<\/strong><\/h4>\n\n\n\n<p>Again, by <strong>Whitney\u2019s inequality<\/strong>: \u03ba(G)\u2264\u03bb(G)\u2264\u03b4(G)\\kappa(G) \\leq \\lambda(G) \\leq \\delta(G)<\/p>\n\n\n\n<p>So if \u03bb(G)=2\\lambda(G) = 2, then <strong>\u03ba(G)\u22642\\kappa(G) \\leq 2<\/strong>. This contradicts the claim \u03ba(G)=3\\kappa(G) = 3. In other words, <strong>the vertex-connectivity of a graph cannot be greater than its edge-connectivity<\/strong>.<\/p>\n\n\n\n<p>This is a strict constraint from graph theory and rules out the possibility of such a graph.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Summary:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>(i)<\/strong>: A graph with \u03ba(G)=2\\kappa(G) = 2, \u03bb(G)=3\\lambda(G) = 3, and \u03b4(G)=4\\delta(G) = 4 <strong>can exist<\/strong>.<\/li>\n\n\n\n<li><strong>(ii)<\/strong>: A graph with \u03ba(G)=3\\kappa(G) = 3, \u03bb(G)=2\\lambda(G) = 2, and \u03b4(G)=4\\delta(G) = 4 <strong>cannot exist<\/strong> due to violation of \u03ba(G)\u2264\u03bb(G)\\kappa(G) \\leq \\lambda(G).<\/li>\n<\/ul>\n\n\n\n<p>We are asked whether there exists a graph GG that satisfies the following properties. Let&#8217;s examine each case:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>i.<\/strong> \u03ba(G)=2\\kappa(G) = 2, \u03bb(G)=3\\lambda(G) = 3, and \u03b4(G)=4\\delta(G) = 4<\/h3>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03ba(G)\\kappa(G): vertex-connectivity \u2013 the minimum number of vertices whose removal disconnects the graph.<\/li>\n\n\n\n<li>\u03bb(G)\\lambda(G): edge-connectivity \u2013 the minimum number of edges whose removal disconnects the graph.<\/li>\n\n\n\n<li>\u03b4(G)\\delta(G): minimum degree \u2013 the smallest degree among all vertices in the graph.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 <strong>Answer (i): Yes, such a graph exists.<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Explanation:<\/strong><\/h4>\n\n\n\n<p>By Whitney&#8217;s inequality: \u03ba(G)\u2264\u03bb(G)\u2264\u03b4(G)\\kappa(G) \\leq \\lambda(G) \\leq \\delta(G)<\/p>\n\n\n\n<p>So if \u03b4(G)=4\\delta(G) = 4, then the edge-connectivity \u03bb(G)\\lambda(G) can be less than 4, and similarly, \u03ba(G)\\kappa(G) can be less than \u03bb(G)\\lambda(G). Therefore, having \u03ba(G)=2\\kappa(G) = 2, \u03bb(G)=3\\lambda(G) = 3, and \u03b4(G)=4\\delta(G) = 4 <strong>does not violate<\/strong> Whitney&#8217;s inequality.<\/p>\n\n\n\n<p><strong>Example Construction:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Take two cliques of 5 vertices (each is K5K_5) with all nodes having degree 4.<\/li>\n\n\n\n<li>Connect the cliques with 3 edges between distinct pairs of nodes (one in each clique).<\/li>\n\n\n\n<li>Remove any two vertices \u2014 the graph remains connected (so \u03ba(G)=2\\kappa(G) = 2).<\/li>\n\n\n\n<li>But removing the three connecting edges disconnects the two K5K_5 parts (so \u03bb(G)=3\\lambda(G) = 3).<\/li>\n\n\n\n<li>Each node in K5K_5 has degree 4, satisfying \u03b4(G)=4\\delta(G) = 4.<\/li>\n<\/ul>\n\n\n\n<p>So such a graph <strong>does exist<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>ii.<\/strong> \u03ba(G)=3\\kappa(G) = 3, \u03bb(G)=2\\lambda(G) = 2, and \u03b4(G)=4\\delta(G) = 4<\/h3>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u274c <strong>Answer (ii): No, such a graph does not exist.<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Explanation:<\/strong><\/h4>\n\n\n\n<p>Again, by <strong>Whitney\u2019s inequality<\/strong>: \u03ba(G)\u2264\u03bb(G)\u2264\u03b4(G)\\kappa(G) \\leq \\lambda(G) \\leq \\delta(G)<\/p>\n\n\n\n<p>So if \u03bb(G)=2\\lambda(G) = 2, then <strong>\u03ba(G)\u22642\\kappa(G) \\leq 2<\/strong>. This contradicts the claim \u03ba(G)=3\\kappa(G) = 3. In other words, <strong>the vertex-connectivity of a graph cannot be greater than its edge-connectivity<\/strong>.<\/p>\n\n\n\n<p>This is a strict constraint from graph theory and rules out the possibility of such a graph.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Summary:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>(i)<\/strong>: A graph with \u03ba(G)=2\\kappa(G) = 2, \u03bb(G)=3\\lambda(G) = 3, and \u03b4(G)=4\\delta(G) = 4 <strong>can exist<\/strong>.<\/li>\n\n\n\n<li><strong>(ii)<\/strong>: A graph with \u03ba(G)=3\\kappa(G) = 3, \u03bb(G)=2\\lambda(G) = 2, and \u03b4(G)=4\\delta(G) = 4 <strong>cannot exist<\/strong> due to violation of \u03ba(G)\u2264\u03bb(G)\\kappa(G) \\leq \\lambda(G).<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Give an example of a graph G with the following properties or explain why no such example exists. Here d(G) denotes the minimum degree of a graph G. i, ii, K(G) = 2, ?(G) &#8211; 3 and d(G) = 4 K(G) = 3, ?(G) = 2 and d(G) = 4 The Correct Answer and Explanation [&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-215745","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/215745","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=215745"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/215745\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=215745"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=215745"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=215745"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}