On Vertex Partitions and the Colin de Verdière Parameter

Abstract We study vertex partitions of graphs according to their Colin de Verdiere parameter μ. By a result of Ding et al. [DOSOO] we know that any graph G with μ ( G ) ⩾ 2 admits a vertex partition into two graphs with μ at most μ ( G ) − 1 . Here we prove that any graph G with μ ( G ) ⩾ 3 admits a vertex partition into three graphs with μ at most μ ( G ) − 2 . This study is extended to other minor-monotone graph parameters like the Hadwiger number.

[1]  László Lovász,et al.  Steinitz Representations of Polyhedra and the Colin de Verdière Number , 2001, J. Comb. Theory, Ser. B.

[2]  Alexander Schrijver,et al.  On a Minor-Monotone Graph Invariant , 1995, J. Comb. Theory, Ser. B.

[3]  L. Lovász,et al.  On the null space of a Colin de Verdière matrix , 1999 .

[4]  Paul D. Seymour,et al.  Graph Minors. XX. Wagner's conjecture , 2004, J. Comb. Theory B.

[5]  Robin Thomas,et al.  Sachs' Linkless Embedding Conjecture , 1995, J. Comb. Theory, Ser. B.

[6]  Y. D. Verdière.  On a novel graph invariant and a planarity criterion , 1990 .

[7]  Leif K. Jørgensen,et al.  Some probabilistic and extremal results on subdivisions and odd subdivisions of graphs , 1989, J. Graph Theory.

[8]  K. Appel,et al.  Every Planar Map Is Four Colorable , 2019, Mathematical Solitaires & Games.

[9]  Bogdan Oporowski,et al.  Surfaces, Tree-Width, Clique-Minors, and Partitions , 2000, J. Comb. Theory, Ser. B.

[10]  K. S. Poh.  On the linear vertex-arboricity of a planar graph , 1990, J. Graph Theory.

[11]  Alexander Schrijver,et al.  A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs , 1998 .

[12]  L. Lovász,et al.  The Colin de Verdière graph parameter , 1999 .

[13]  G. Chartrand,et al.  Graphs with Forbidden Subgraphs , 1971 .

[14]  Wayne Goddard,et al.  Acyclic colorings of planar graphs , 1991, Discret. Math..

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

[16]  K. Wagner.  Beweis einer Abschwächung der Hadwiger-Vermutung , 1964 .