TY - JOUR
T1 - Subdivisions in apex graphs
AU - Aigner-Horev, Elad
PY - 2012/4
Y1 - 2012/4
N2 - The Kelmans-Seymour conjecture states that every 5-connected nonplanar graph contains a subdivided K 5. Certain questions of Mader propose a "plan" towards a possible resolution of this conjecture. One part of this plan is to show that every 5-connected nonplanar graph containing K 4 - or K 2,3 as a subgraph has a subdivided K 5. Recently, Ma and Yu showed that every 5-connected nonplanar graph containing K 4 - as a subgraph has a subdivided K 5. We take interest in K 2,3 and prove that every 5-connected nonplanar apex graph containing K 2,3 as a subgraph contains a subdivided K 5. The result of Ma and Yu can be used in a short discharging argument to prove that every 5-connected nonplanar apex graph contains a subdivided K 5; here we propose a longer proof whose merit is that it avoids the use of discharging and employs a more structural approach; consequently it is more amenable to generalization.
AB - The Kelmans-Seymour conjecture states that every 5-connected nonplanar graph contains a subdivided K 5. Certain questions of Mader propose a "plan" towards a possible resolution of this conjecture. One part of this plan is to show that every 5-connected nonplanar graph containing K 4 - or K 2,3 as a subgraph has a subdivided K 5. Recently, Ma and Yu showed that every 5-connected nonplanar graph containing K 4 - as a subgraph has a subdivided K 5. We take interest in K 2,3 and prove that every 5-connected nonplanar apex graph containing K 2,3 as a subgraph contains a subdivided K 5. The result of Ma and Yu can be used in a short discharging argument to prove that every 5-connected nonplanar apex graph contains a subdivided K 5; here we propose a longer proof whose merit is that it avoids the use of discharging and employs a more structural approach; consequently it is more amenable to generalization.
KW - (Rooted) subdivisions
KW - Apex graphs
KW - Nonseparating paths
UR - http://www.scopus.com/inward/record.url?scp=84861226664&partnerID=8YFLogxK
U2 - 10.1007/s12188-012-0063-x
DO - 10.1007/s12188-012-0063-x
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AN - SCOPUS:84861226664
SN - 0025-5858
VL - 82
SP - 83
EP - 113
JO - Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
JF - Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
IS - 1
ER -