{"id":4851,"date":"2023-04-19T08:36:56","date_gmt":"2023-04-19T12:36:56","guid":{"rendered":"http:\/\/matroidunion.org\/?p=4851"},"modified":"2023-04-27T18:49:43","modified_gmt":"2023-04-27T22:49:43","slug":"online-talk-sebastian-mies","status":"publish","type":"post","link":"https:\/\/matroidunion.org\/?p=4851","title":{"rendered":"Online Talk: Sebastian Mies"},"content":{"rendered":"\n<p><strong>YouTube recording:<\/strong> <a href=\"https:\/\/www.youtube.com\/watch?v=3KYEAEiHkb0\" target=\"_blank\" rel=\"noopener\">https:\/\/www.youtube.com\/watch?v=3KYEAEiHkb0<\/a><\/p>\n<p><strong>Time: <\/strong>Thursday, Apr 27, 3pm ET<br \/><strong>Zoom: <\/strong><a href=\"https:\/\/gatech.zoom.us\/j\/8802082683\" target=\"_blank\" rel=\"noopener\">https:\/\/gatech.zoom.us\/j\/8802082683<\/a><strong><br \/><\/strong><br \/><strong>Speaker:\u00a0<\/strong>Sebastian Mies, Johannes Gutenberg University Mainz<br \/><strong>Title: <\/strong>The Strong Nine Dragon Tree Conjecture for $d \\le k+1$<br \/><br \/><strong>Abstract: <\/strong>The arboricity $\\Gamma(G)$ of an undirected graph $G = (V,E)$ is the minimal number such that $E$ can be partitioned into $\\Gamma(G)$ forests. Nash-Williams&#8217; formula states that $\\Gamma(G) = \\lceil \\gamma(G) \\rceil$, where $\\gamma(G)$ is the maximum of $|E_H|\/(|V_H|-1)$ over all subgraphs $(V_H, E_H)$ of $G$ with $|V_H| \\ge 2$.<br aria-hidden=\"true\" \/><br aria-hidden=\"true\" \/>The Strong Nine Dragon Tree Conjecture states that if $\\gamma(G) \\le k + d \/ (d+k+1)$ for natural numbers k, d, then there is a partition of the edge set of G into k+1 forests such that one forest has at most d edges in each connected component.<br aria-hidden=\"true\" \/><br aria-hidden=\"true\" \/>We settle the conjecture for $d \\le k + 1$. For $d \\le 2(k+1)$, we cannot prove the conjecture, however we show that there exists a partition in which the connected components in one forest have at most $d + \\lceil kd\/(k+1) \\rceil &#8211; k$ edges.<br aria-hidden=\"true\" \/><br aria-hidden=\"true\" \/>As an application of this theorem, we show that every 5-edge-connected planar graph G has a 5\/6-thin spanning tree, a spanning tree whose edges fill up at most 5\/6 of every cut. This theorem is best possible, in the sense that we cannot replace 5-edge-connected with 4-edge-connected, even if we replace 5\/6 with any positive real number less than 1. This strengthens a result of Merker and Postle which showed 6-edge-connected planar graphs have a 18\/19-thin spanning tree.<br aria-hidden=\"true\" \/>This is joint work with Benjamin Moore.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>YouTube recording: https:\/\/www.youtube.com\/watch?v=3KYEAEiHkb0 Time: Thursday, Apr 27, 3pm ETZoom: https:\/\/gatech.zoom.us\/j\/8802082683Speaker:\u00a0Sebastian Mies, Johannes Gutenberg University MainzTitle: The Strong Nine Dragon Tree Conjecture for $d \\le k+1$ Abstract: The arboricity $\\Gamma(G)$ of an undirected graph $G = (V,E)$ is the minimal number &hellip; <a href=\"https:\/\/matroidunion.org\/?p=4851\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":21,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[11],"class_list":["post-4851","post","type-post","status-publish","format-standard","hentry","category-matroids","tag-online-talks"],"_links":{"self":[{"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/posts\/4851","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/users\/21"}],"replies":[{"embeddable":true,"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4851"}],"version-history":[{"count":7,"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/posts\/4851\/revisions"}],"predecessor-version":[{"id":4862,"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/posts\/4851\/revisions\/4862"}],"wp:attachment":[{"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4851"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4851"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4851"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}