Subdivisions of K5 in graphs containing K2, 3

Seymour and, independently, Kelmans conjectured that every 5-connected nonplanar graph contains a subdivision of K 5 . We prove this conjecture for graphs containing K 2 , 3 . As a consequence, the Kelmans-Seymour conjecture is true if the answer to the following question of Mader is affirmative: Does every simple graph on n ? 4 vertices with more than 12 ( n - 2 ) / 5 edges contain a K 4 - , a K 2 , 3 , or a subdivision of K 5 ?

[1]  André E. Kézdy,et al.  Do 3n - 5 edges force a subdivision of K5? , 1991, J. Graph Theory.

[2]  Wolfgang Mader,et al.  3n − 5 Edges Do Force a Subdivision of , 1998, Comb..

[3]  G. Dirac.  A Property of 4-Chromatic Graphs and some Remarks on Critical Graphs , 1952 .

[4]  Yossi Shiloach,et al.  A Polynomial Solution to the Undirected Two Paths Problem , 1980, JACM.

[5]  Xingxing Yu,et al.  Nonseparating Cycles in 4-Connected Graphs , 2003, SIAM J. Discret. Math..

[6]  Xingxing Yu.  Subdivisions in Planar Graphs , 1998, J. Comb. Theory, Ser. B.

[7]  Carsten Thomassen,et al.  2-Linked Graphs , 1980, Eur. J. Comb..

[8]  Xingxing Yu,et al.  Nonseparating Planar Chains in 4-Connected Graphs , 2005, SIAM J. Discret. Math..

[9]  Jie Ma,et al.  Independent paths and K5-subdivisions , 2010, J. Comb. Theory, Ser. B.

[10]  Jie Ma,et al.  K5-Subdivisions in graphs containing K-4 , 2013, J. Comb. Theory, Ser. B.

[11]  Hazel Perfect,et al.  Applications of Menger's graph theorem , 1968 .

[12]  M. Watkins,et al.  Cycles and Connectivity in Graphs , 1967, Canadian Journal of Mathematics.

[13]  Neil Robertson,et al.  Covering Three Edges with a Bond in a Nonseparable Graph , 1980 .

[14]  W. T. Tutte.  How to Draw a Graph , 1963 .

[15]  Roi Krakovski,et al.  Subdivisions of K5 in Graphs Embedded on Surfaces With Face-Width at Least 5 , 2013, J. Graph Theory.

[16]  Elad Aigner-Horev.  Subdivisions in apex graphs , 2010 .

[17]  Paul D. Seymour.  Disjoint paths in graphs , 2006, Discret. Math..