{"id":758,"date":"2014-06-02T11:39:15","date_gmt":"2014-06-02T15:39:15","guid":{"rendered":"http:\/\/matroidunion.org\/?p=758"},"modified":"2014-06-02T11:39:53","modified_gmt":"2014-06-02T15:39:53","slug":"a-question-from-jim-geelen","status":"publish","type":"post","link":"https:\/\/matroidunion.org\/?p=758","title":{"rendered":"A question from Jim Geelen about matroid circuits"},"content":{"rendered":"<p><i>Guest post by <a href=\"http:\/\/www.math.uwaterloo.ca\/~jfgeelen\/\">Jim Geelen<\/a><\/i> $\\newcommand{\\del}{\\,\\backslash\\,}$ $\\newcommand{\\con}{\/}$ $\\newcommand{\\bF}{\\mathbb{F}}$ $\\newcommand{\\cL}{\\mathcal{L}}$ $\\newcommand{\\lt}{&#060;}$ $\\newcommand{\\gt}{&#062;}$ $\\DeclareMathOperator{\\ex}{ex}$ $\\DeclareMathOperator{\\rank}{rank}$ $\\DeclareMathOperator{\\PG}{PG}$ $\\DeclareMathOperator{\\AG}{AG}$ $\\DeclareMathOperator{\\BB}{BB}$<\/p>\n<p>I was asked a question recently, by Anna Lubiw and Vinayak Pathak, which led me to the following conjecture. It looks quite natural, so I wonder whether it is already known.<\/p>\n<p><strong> Conjecture: <\/strong><i>In any rank-$r$ matroid $M$, that has a circuit, there is a circuit $C$ such that each component of $M\\con C$ has rank at most $r\/2$.<\/i><\/p>\n<p>I would be happy with $r\/2$ replaced by $\\alpha r$ for any $\\alpha \\lt 1$. The conjecture holds for graphic and cographic matroids.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Guest post by Jim Geelen $\\newcommand{\\del}{\\,\\backslash\\,}$ $\\newcommand{\\con}{\/}$ $\\newcommand{\\bF}{\\mathbb{F}}$ $\\newcommand{\\cL}{\\mathcal{L}}$ $\\newcommand{\\lt}{&#060;}$ $\\newcommand{\\gt}{&#062;}$ $\\DeclareMathOperator{\\ex}{ex}$ $\\DeclareMathOperator{\\rank}{rank}$ $\\DeclareMathOperator{\\PG}{PG}$ $\\DeclareMathOperator{\\AG}{AG}$ $\\DeclareMathOperator{\\BB}{BB}$ I was asked a question recently, by Anna Lubiw and Vinayak Pathak, which led me to the following conjecture. It looks quite natural, so &hellip; <a href=\"https:\/\/matroidunion.org\/?p=758\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-758","post","type-post","status-publish","format-standard","hentry","category-matroids"],"_links":{"self":[{"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/posts\/758","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\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=758"}],"version-history":[{"count":3,"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/posts\/758\/revisions"}],"predecessor-version":[{"id":762,"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/posts\/758\/revisions\/762"}],"wp:attachment":[{"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=758"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=758"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=758"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}