Maximum number of colourings: 4-chromatic graphs

Abstract It is proved that every connected graph G on n vertices with χ ( G ) ≥ 4 has at most k ( k − 1 ) n − 3 ( k − 2 ) ( k − 3 ) k-colourings for every k ≥ 4 . Equality holds for some (and then for every) k if and only if the graph is formed from K 4 by repeatedly adding leaves. This confirms (a strengthening of) the 4-chromatic case of a long-standing conjecture of Tomescu [29] . Proof methods may be of independent interest. In particular, one of our auxiliary results about list-chromatic polynomials solves a generalized version of a recent conjecture of Brown, Erey, and Li.

[1]  George D. Birkhoff,et al.  On the Number of Ways of Colouring a Map , 1930 .

[2]  I. Agol,et al.  Tutte relations, TQFT, and planarity of cubic graphs , 2015, 1512.07339.

[3]  Jason I. Brown,et al.  Extremal Restraints for Graph Colourings , 2016, 1611.08920.

[4]  Jie Ma,et al.  Maximizing proper colorings on graphs , 2014, J. Comb. Theory, Ser. B.

[5]  Aysel Erey,et al.  Maximizing the number of x-colorings of 4-chromatic graphs , 2016, Discret. Math..

[6]  Jason I. Brown,et al.  New bounds for chromatic polynomials and chromatic roots , 2015, Discret. Math..

[7]  Felix Lazebnik,et al.  Some corollaries of a theorem of Whitney on the chromatic polynomial , 1991, Discret. Math..

[8]  Klaus Dohmen.  Bounds to the Chromatic Polynomial of a Graph , 1998 .

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

[10]  Italo Simonelli,et al.  Optimal graphs for chromatic polynomials , 2008, Discret. Math..

[11]  Felix Lazebnik,et al.  New upper bounds for the greatest number of proper colorings of a (V, E)-graph , 1990, J. Graph Theory.

[12]  Klaus Dohmen,et al.  Lower bounds and upper bounds for chromatic polynomials , 1993, J. Graph Theory.

[13]  Ioan Tomescu,et al.  Maximum chromatic polynomials of 2-connected graphs , 1994, J. Graph Theory.

[14]  G. Birkhoff.  A Determinant Formula for the Number of Ways of Coloring a Map , 1912 .

[15]  Felix Lazebnik,et al.  The maximum number of colorings of graphs of given order and size: A survey , 2019, Discret. Math..

[16]  Herbert S. Wilf,et al.  Backtrack: An O(1) Expected Time Algorithm for the Graph Coloring Problem , 1984, Inf. Process. Lett..

[17]  Nathan Linial.  Legal coloring of graphs , 1986, Comb..

[18]  O. Pikhurko,et al.  Maximizing the number of q‐colorings , 2008, 0811.2625.

[19]  F. Lazebnik,et al.  Maximum number of colorings of (2k, k 2 )-graphs , 2007 .

[20]  László Lovász,et al.  Large Networks and Graph Limits , 2012, Colloquium Publications.

[21]  Li Zhang,et al.  The minimum number of vertices for a triangle-free graph with chil(G)=4 is 11 , 2008, Discret. Math..

[22]  Ioan Tomescu,et al.  Maximum chromatic polynomial of 3-chromatic blocks , 1997, Discret. Math..

[23]  Bojan Mohar,et al.  Maximum Number of Colourings: 5-Chromatic Case , 2019, Electron. J. Comb..

[24]  Edward A. Bender,et al.  A Theoretical Analysis of Backtracking in the Graph Coloring Problem , 1985, J. Algorithms.

[25]  K. Koh,et al.  Chromatic polynomials and chro-maticity of graphs , 2005 .

[26]  John Engbers,et al.  Extremal H‐Colorings of Graphs with Fixed Minimum Degree , 2013, J. Graph Theory.

[27]  Ioan Tomescu,et al.  Maximal chromatic polynomials of connected planar graphs , 1990, J. Graph Theory.

[28]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[29]  Jason I. Brown,et al.  Restraints permitting the largest number of colourings , 2017, Discret. Appl. Math..

[30]  David Galvin,et al.  Extremal H-colorings of trees and 2-connected graphs , 2015, J. Comb. Theory, Ser. B.

[31]  Jacob Fox,et al.  A proof of Tomescu's graph coloring conjecture , 2017, J. Comb. Theory, Ser. B.