More results on r-inflated graphs: Arboricity, thickness, chromatic number and fractional chromatic number

The r -inflation of a graph G is the lexicographic product G with K r . A graph is said to have thickness t if its edges can be partitioned into t sets, each of which induces a planar graph, and t is smallest possible. In the setting of the r -inflation of planar graphs, we investigate the generalization of Ringel's famous Earth-Moon problem: What is the largest chromatic number of any thickness-t graph? In particular, we study classes of planar graphs for which we can determine both the thickness and chromatic number of their 2-inflations, and provide bounds on these parameters for their r -inflations. Moreover, in the same setting, we investigate arboricity and fractional chromatic number as well. We end with a list of open questions.

[1]  M. Ziegler.  Volume 152 of Graduate Texts in Mathematics , 1995 .

[2]  J. Schiff.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[3]  E. Scheinerman,et al.  Fractional Graph Theory: A Rational Approach to the Theory of Graphs , 1997 .

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

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

[6]  Jorge Nuno Silva,et al.  Mathematical Games , 1959, Nature.

[7]  B. M. Fulk.  MATH , 1992 .

[8]  Martin Gardner,et al.  The last recreations : hydras, eggs, and other mathematical mystifications , 1997 .

[9]  Ellen Gethner,et al.  Thickness‐two graphs part one: New nine‐critical graphs, permuted layer graphs, and Catlin's graphs , 2008, J. Graph Theory.

[10]  Brad Jackson,et al.  Maps of m-pires on the projective plane , 1983, Discret. Math..

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[13]  Paul D. Seymour,et al.  Graph Minors. XIX. Well-quasi-ordering on a surface , 2004, J. Comb. Theory, Ser. B.

[14]  Carsten Thomassen,et al.  Some remarks on Hajo's' conjecture , 2005, J. Comb. Theory, Ser. B.

[15]  Xuding Zhu,et al.  Star-extremal graphs and the lexicographic product , 1996, Discret. Math..

[16]  Michael Stiebitz,et al.  On a special case of Hadwiger's conjecture , 2003, Discuss. Math. Graph Theory.

[17]  J. Moon,et al.  On the thickness of the complete bipartite graph , 1964, Mathematical Proceedings of the Cambridge Philosophical Society.

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

[19]  N. M. Korneyenko.  Combinatorial Algorithms on a Class of Graphs , 1994, Discret. Appl. Math..

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

[21]  V B Alekseev,et al.  THE THICKNESS OF AN ARBITRARY COMPLETE GRAPH , 1976 .

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

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

[24]  A. Kempe.  On the Geographical Problem of the Four Colours , 1879 .

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

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

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

[28]  Brian Alspach,et al.  Isomorphism of circulant graphs and digraphs , 1979, Discret. Math..

[29]  Robin J. Wilson.  Four Colors Suffice: How the Map Problem Was Solved , 2002 .

[30]  Paul C. Kainen,et al.  Some recent results in topological graph theory , 1974 .

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

[32]  P. J. Heawood.  Map-Colour Theorem , 1949 .

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

[34]  Ellen Gethner,et al.  Thickness-Two Graphs Part Two: More New Nine-Critical Graphs, Independence Ratio, Cloned Planar Graphs, and Singly and Doubly Outerplanar Graphs , 2009, Graphs Comb..

[35]  Martin Gardner,et al.  The Last Recreations , 1997 .

[36]  Brad Jackson,et al.  Variations on Ringel's earth-moon problem , 2000, Discret. Math..

[37]  B. Elspas,et al.  Graphs with circulant adjacency matrices , 1970 .

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

[39]  G. Ringel.  Färbungsprobleme auf Flächen und Graphen , 1959 .

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

[41]  Ellen Gethner,et al.  The thickness and chromatic number of r-inflated graphs , 2010, Discret. Math..

[42]  R. Duffin.  Topology of series-parallel networks , 1965 .

[43]  David R. Wood,et al.  Graph Treewidth and Geometric Thickness Parameters , 2005, GD.

[44]  Clemens Heuberger,et al.  On planarity and colorability of circulant graphs , 2003, Discret. Math..