Thickness-Two Graphs Part Two: More New Nine-Critical Graphs, Independence Ratio, Cloned Planar Graphs, and Singly and Doubly Outerplanar Graphs

There are numerous means for measuring the closeness to planarity of a graph such as crossing number, splitting number, and a variety of thickness parameters. We focus on the classical concept of the thickness of a graph, and we add to earlier work in [4]. In particular, we offer new 9-critical thickness-two graphs on 17, 25, and 33 vertices, all of which provide counterexamples to a conjecture on independence ratio of Albertson; we investigate three classes of graphs, namely singly and doubly outerplanar graphs, and cloned planar graphs. We give a sharp upper bound for the largest chromatic number of the cloned planar graphs, and we give upper and lower bounds for the largest chromatic number of the former two classes.

[1]  D. West.  Introduction to Graph Theory , 1995 .

[2]  R. Guy,et al.  The Outerthickness & Outercoarseness of Graphs I. The Complete Graph & The n-Cube , 1990 .

[3]  K. Appel,et al.  Microfiche supplement to “Every planar map is four colorable” , 1977 .

[4]  G. Koester.  4-critical 4-valent planar graphs constructed with crowns. , 1990 .

[5]  Debra L. Boutin,et al.  Thickness-two graphs part one: New nine-critical graphs, permuted layer graphs, and Catlin's graphs , 2008 .

[6]  David S. Johnson,et al.  Some simplified NP-complete problems , 1974, STOC '74.

[7]  Samir Khuller,et al.  Four colors suffice! , 2005, SIGA.

[8]  W. T. Tutte.  The Non-Biplanar Character of the Complete 9-Graph , 1963, Canadian Mathematical Bulletin.

[9]  J. Hutchinson,et al.  Coloring Ordinary Maps, Maps of Empires, and Maps of the Moon , 1993 .

[10]  Anthony Mansfield,et al.  Determining the thickness of graphs is NP-hard , 1983, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  K. Appel,et al.  Every planar map is four colorable. Part II: Reducibility , 1977 .

[12]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

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

[14]  P. Seymour,et al.  A new proof of the four-colour theorem , 1996 .

[15]  Donald L. Kreher,et al.  Graphs, algorithms and optimization , 2004 .

[16]  K. Appel,et al.  Every planar map is four colorable. Part I: Discharging , 1977 .

[18]  F. Harary,et al.  Every planar graph with nine points has a nonplanar complement , 1962 .