Independent paths and K5-subdivisions

A well-known theorem of Kuratowski states that a graph is planar iff it contains no subdivision of K"5 or K"3","3. Seymour conjectured in 1977 that every 5-connected nonplanar graph contains a subdivision of K"5. In this paper, we prove several results about independent paths (no vertex of a path is internal to another), which are then used to prove Seymour's conjecture for two classes of graphs. These results will be used in a subsequent paper to prove Seymour's conjecture for graphs containing K"4^-, which is a step in a program to approach Seymour's conjecture.

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

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

[3]  Reinhard Diestel,et al.  Graph Theory , 1997 .

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

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

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

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

[8]  Xingxing Yu,et al.  Graph Connectivity After Path Removal , 2003, Comb..

[9]  Ken-ichi Kawarabayashi,et al.  Non-Separating Paths in 4-Connected Graphs , 2005 .

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

[11]  Ken-ichi Kawarabayashi,et al.  Contractible Edges and Triangles in k-Connected Graphs , 2002, J. Comb. Theory, Ser. B.

[12]  Ken-ichi Kawarabayashi.  Note on k-contractible edges in k-connected graphs , 2001, Australas. J Comb..

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

[14]  Paul A. Catlin,et al.  Hajós' graph-coloring conjecture: Variations and counterexamples , 1979, J. Comb. Theory, Ser. B.

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

[16]  Xingxing Yu,et al.  Reducing Hajós' 4-coloring conjecture to 4-connected graphs , 2006, J. Comb. Theory, Ser. B.

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

[18]  Matthias Kriesell.  Induced paths in 5-connected graphs , 2001 .

[19]  Robin Thomas,et al.  Hadwiger's conjecture forK6-free graphs , 1993, Comb..

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