DP-Coloring Cartesian Products of Graphs

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvořák and Postle in 2015. Motivated by results related to list coloring Cartesian products of graphs, we initiate the study of the DP-chromatic number, χDP , of the same. We show that χDP (G H) ≤ min{χDP (G) + col(H), χDP (H) + col(G)} − 1 where col(H) is the coloring number of the graph H . We focus on building tools for lower bound arguments for χDP (G H) and use them to show the sharpness of the bound above and its various forms. Our results illustrate that the DP color function of G, the DP analogue of the chromatic polynomial, is essential in the study of the DP-chromatic number of the Cartesian product of graphs, including the following question that extends the sharpness problem above and the classical result on gap between list chromatic number and chromatic number: given any graph G and k ∈ N, what is the smallest t for which χDP (G Kk,t) = χDP (G) + k?

[1]  Anton Bernshteyn,et al.  The asymptotic behavior of the correspondence chromatic number , 2016, Discret. Math..

[2]  Quentin Donner,et al.  On the number of list-colorings , 1992, J. Graph Theory.

[3]  Ramin Naimi,et al.  List Coloring And n-Monophilic Graphs , 2010, Ars Comb..

[4]  Jeffrey A. Mudrock.  A note on the DP-chromatic number of complete bipartite graphs , 2018, Discret. Math..

[5]  Wilfried Imrich,et al.  Recognizing Cartesian products in linear time , 2007, Discret. Math..

[6]  Jeffrey A. Mudrock,et al.  Answers to Two Questions on the DP Color Function , 2020, Electron. J. Comb..

[7]  Stanislav Jendrol',et al.  List coloring of Cartesian products of graphs , 2006, Discret. Math..

[8]  Luke Postle,et al.  Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8 , 2015, J. Comb. Theory B.

[9]  Yan Yang,et al.  Compare DP color functions with chromatic polynomials , 2021, 2105.11081.

[10]  Hemanshu Kaul,et al.  On the Chromatic Polynomial and Counting DP-Colorings , 2019, Adv. Appl. Math..

[11]  Jeffrey A. Mudrock,et al.  On Polynomial Representations of the DP Color Function: Theta Graphs and Their Generalizations , 2020, 2012.12897.

[12]  Wei Wang,et al.  When does the list-coloring function of a graph equal its chromatic polynomial , 2017, J. Comb. Theory, Ser. B.

[13]  Alexandr Kostochka,et al.  On Differences Between DP-Coloring and List Coloring , 2017, Siberian Advances in Mathematics.

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

[15]  Jeffrey A. Mudrock.  A Deletion–Contraction Relation for the DP Color Function , 2021, Graphs and Combinatorics.

[16]  Carsten Thomassen,et al.  The chromatic polynomial and list colorings , 2009, J. Comb. Theory, Ser. B.

[17]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[18]  Hemanshu Kaul,et al.  List Coloring a Cartesian Product with a Complete Bipartite Factor , 2018, Graphs and Combinatorics.

[19]  Gert Sabidussi,et al.  Graph multiplication , 1959 .

[20]  Hemanshu Kaul,et al.  The DP Color Function of Joins and Vertex-Gluings of Graphs , 2021 .