Subdivisions of K5 in Graphs Embedded on Surfaces With Face‐Width at Least 5. Issue 2 (24th October 2012)

• Record Type: Journal Article
• Authors: Krakovski, Roi; Stephens, D. Christopher; Zha, Xiaoya
• Is Part Of: Journal of graph theory. Volume 74:Issue 2(2013)
• Page Start: 182
• Page End: 197
• Publication Date: 2012-10-24
• Subjects: Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.21700
• Languages: English
• ISSNs: 0364-9024

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We prove that if <italic>G</italic> is a 5‐connected graph embedded on a surface Σ (other than the sphere) with face‐width at least 5, then <italic>G</italic> contains a subdivision of <italic>K</italic>₅. This is a special case of a conjecture of P. Seymour, that every 5‐connected nonplanar graph contains a subdivision of <italic>K</italic>₅. Moreover, we prove that if <italic>G</italic> is 6‐connected and embedded with face‐width at least 5, then for every <italic>v</italic> ∈ <italic>V</italic>(G), <italic>G</italic> contains a subdivision of <italic>K</italic>₅ whose branch vertices are <italic>v</italic> and four neighbors of <italic>v</italic>.</p> </abstract>