Surfaces, Tree-Width, Clique-Minors, and Partitions

In 1971, Chartrand, Geller, and Hedetniemi conjectured that the edge set of a planar graph may be partitioned into two subsets, each of which induces an outerplanar graph. Some partial results towards this conjecture are presented. One such result, in which a planar graph may be thus edge partitioned into two series-parallel graphs, has nice generalizations for graphs embedded onto an arbitrary surface and graphs with no large clique-minor. Several open questions are raised.

[1]  C. Nash-Williams.  Decomposition of Finite Graphs Into Forests , 1964 .

[2]  A. Thomason.  An extremal function for contractions of graphs , 1984, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  Yue Zhao,et al.  On 2-Connected Spanning Subgraphs with Low Maximum Degree , 1998, J. Comb. Theory, Ser. B.

[4]  Paul D. Seymour,et al.  Graph minors. IV. Tree-width and well-quasi-ordering , 1990, J. Comb. Theory, Ser. B.

[5]  Daniel P. Sanders.  Linear algorithms for graphs of tree-width at most four , 1993 .

[6]  W. T. Tutte.  A THEOREM ON PLANAR GRAPHS , 1956 .

[7]  W. Mader.  Homomorphiesätze für Graphen , 1968 .

[8]  Gary Chartrand,et al.  The point-arboricity of a graph , 1968 .

[9]  Paul D. Seymour,et al.  Graph minors. III. Planar tree-width , 1984, J. Comb. Theory B.

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

[11]  A. Satyanarayana,et al.  A characterization of partial 3-trees , 1990, Networks.

[12]  Neil Robertson,et al.  Computing the orientable genus of projective graphs , 1995, J. Graph Theory.

[13]  Robin Thomas,et al.  The Four-Colour Theorem , 1997, J. Comb. Theory, Ser. B.

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

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

[16]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[17]  Daniel P. Sanders.  On Linear Recognition of Tree-Width at Most Four , 1996, SIAM J. Discret. Math..

[18]  Stefan Arnborg,et al.  Forbidden minors characterization of partial 3-trees , 1990, Discret. Math..

[19]  Tommy R. Jensen,et al.  Graph Coloring Problems , 1994 .

[20]  Oleg V. Borodin.  On acyclic colorings of planar graphs , 1979, Discret. Math..

[21]  S. Arnborg,et al.  Characterization and recognition of partial 3-trees , 1986 .

[22]  Charles J. Colbourn,et al.  Partitioning the Edges of a Planar Graph into Two Partial K-Trees , 1988 .

[23]  Daniel P. Sanders.  On Hamilton cycles in certain planar graphs , 1996, J. Graph Theory.

[24]  Stefan Arnborg,et al.  Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey , 1985, BIT.

[25]  Kiran S. Kedlaya.  Outerplanar Partitions of Planar Graphs , 1996, J. Comb. Theory, Ser. B.

[26]  Leif K. Jørgensen,et al.  Contractions to k8 , 1994, J. Graph Theory.

[27]  G. Chartrand,et al.  The Point‐Arboricity of Planar Graphs , 1969 .

[28]  Robin J. Wilson.  EVERY PLANAR MAP IS FOUR COLORABLE , 1991 .