Graphs and Combinatorics

, Volume 35, Issue 6, pp 1571–1583 | Cite as

List Coloring a Cartesian Product with a Complete Bipartite Factor

  • Hemanshu KaulEmail author
  • Jeffrey A. Mudrock
Original Paper


We study the list chromatic number of the Cartesian product of any graph G and a complete bipartite graph with partite sets of size a and b, denoted \(\chi _\ell (G \square K_{a,b})\). We have two motivations. A classic result on the gap between list chromatic number and the chromatic number tells us \(\chi _\ell (K_{a,b}) = 1 + a\) if and only if \(b \ge a^a\). Since \(\chi _\ell (K_{a,b}) \le 1 + a\) for any \(b \in {\mathbb {N}}\), this result tells us the values of b for which \(\chi _\ell (K_{a,b})\) is as large as possible and far from \(\chi (K_{a,b})=2\). In this paper we seek to understand when \(\chi _\ell (G \square K_{a,b})\) is far from \(\chi (G \square K_{a,b}) = \max \{\chi (G), 2 \}\). It is easy to show \(\chi _\ell (G \square K_{a,b}) \le \chi _\ell (G) + a\). Borowiecki et al. (Discrete Math 306:1955–1958, 2006) showed that this bound is attainable if b is sufficiently large; specifically, \(\chi _\ell (G \square K_{a,b}) = \chi _\ell (G) + a\) whenever \(b \ge (\chi _\ell (G) + a - 1)^{a|V(G)|}\). Given any graph G and \(a \in {\mathbb {N}}\), we wish to determine the smallest b such that \(\chi _\ell (G \square K_{a,b}) = \chi _\ell (G) + a\). In this paper we show that the list color function, a list analogue of the chromatic polynomial, provides the right concept and tool for making progress on this problem. Using the list color function, we prove a general improvement on Borowiecki et al.’s (2006) result, and we compute the smallest such b for some large families of chromatic-choosable graphs.


Graph coloring List coloring Cartesian product List color function Chromatic choosable 

Mathematics Subject Classification




The authors would like to thank the two anonymous referees whose comments helped improve the readability of this note.


  1. 1.
    Birkhoff, G.D.: A determinant formula for the number of ways of coloring a map. Ann. Math. 14, 42–46 (1912)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Borodin, O.V., Kostochka, A.V., Woodall, D.R.: List edge and list total colourings of multigraphs. J. Comb. Theory Ser. B 71(2), 184–204 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Borowiecki, M., Jendrol, S., Král, D., Miškuf, J.: List coloring of cartesian products of graphs. Discrete Math. 306, 1955–1958 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Erdős, P., Rubin, A.L., Taylor, H.: Choosability in graphs. Congr. Numer. 26, 125–127 (1979)zbMATHGoogle Scholar
  5. 5.
    Galvin, F.: The list chromatic index of a bipartite multigraph. J. Comb. Theory Ser. B 63(1), 153–158 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gravier, S., Maffray, F.: Choice number of 3-colorable elementary graphs. Discrete Math. 165/166, 353–358 (1997). (Graphs and combinatorics (Marseille, 1995)) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Häggkvist, R., Chetwynd, A.: Some upper bounds on the total and list chromatic numbers of multigraphs. J. Graph Theory 16(5), 503–516 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kaul, H., Mudrock, J.: Criticality, the list color function, and list coloring the Cartesian product of graphs (2018). arXiv:1805.02147
  9. 9.
    Kirov, R., Naimi, R.: List coloring and \(n\)-monophilic graphs. Ars Comb. 124, 329–340 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kostochka, A.V., Sidorenko, A.: Problem Session. Fourth Czechoslovak Symposium on Combinatorics. Prachatice, Juin (1990)Google Scholar
  11. 11.
    Noel, J.A., Reed, B.A., Wu, H.: A proof of a conjecture of ohba. J. Graph Theory 79(2), 86–102 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ohba, K.: On chromatic-choosable graphs. J. Graph Theory 40(2), 130–135 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Prowse, A., Woodall, D.R.: Choosability of powers of circuits. Graphs Comb. 19, 137–144 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Read, R.C.: An introduction to chromatic polynomials. J. Comb. Theory 4, 52–71 (1968)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Thomassen, C.: The chromatic polynomial and list colorings. J. Comb. Theory Ser. B 99, 474–479 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Tuza, Zs, Voigt, M.: On a conjecture of Erdős, Rubin, and Taylor. Tatra Mt. Math. Publ. 9, 69–82 (1996)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Vizing, V.G.: Coloring the vertices of a graph in prescribed colors. Metody Diskret. Anal. v Teorii Kodovi Skhem 101, 3–10 (1976). (Diskret. Analiz. no. 29) MathSciNetGoogle Scholar
  18. 18.
    Wang, W., Qian, J., Yan, Z.: When does the list-coloring function of a graph equal its chromatic polynomial. J. Comb. Theory Ser. B 122, 543–549 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    West, D.B.: Introduction to Graph Theory. Prentice Hall, Upper Saddle River (2001)Google Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA
  2. 2.Department of MathematicsCollege of Lake CountyGrayslakeUSA

Personalised recommendations