On Properly Ordered Coloring of Vertices in a Vertex-Weighted Graph


We introduce the notion of a properly ordered coloring (POC) of a weighted graph, that generalizes the notion of vertex coloring of a graph. Under a POC, if xy is an edge, then the larger weighted vertex receives a larger color; in the case of equal weights of x and y, their colors must be different. In this paper, we shall initiate the study of this special coloring in graphs. For a graph G, we introduce the function f(G) which gives the maximum number of colors required by a POC over all weightings of G. We show that f(G) = (G), where (G) is the number of vertices of a longest path in G. Another function we introduce is χPOC(G; t) giving the minimum number of colors required over all weightings of G using t distinct weights. We show that the ratio of χPOC(G; t) − 1 to χ(G) − 1 can be bounded by t for any graph G; in fact, the result is shown by determining χPOC(G; t) when G is a complete multipartite graph. We also determine the minimum number of colors to give a POC on a vertex-weighted graph in terms of the number of vertices of a longest directed path in an orientation of the underlying graph. This extends the so called Gallai-Hasse-Roy-Vitaver theorem, a classical result concerning the relationship between the chromatic number of a graph G and the number of vertices of a longest directed path in an orientation of G.

This is a preview of subscription content, access via your institution.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.


  1. 1.

    Chaitin, G.J.: Register allocation & spilling via graph colouring. In: Proceedings of the 1982 SIGPLAN Symposium on Compiler Construction, pp 98–105 (1982)

  2. 2.

    Gallai, T.: On directed graphs and circuits. In: Theory of Graphs (Proceedings of the Colloquium held at Tihany 1966), pp 115–118 (1968)

  3. 3.

    Hasse, M.: Zur algebraischen Begründung der Graphentheorie. I. Math. Nachr. (in German) 28, 275–290 (1965)

    Article  Google Scholar 

  4. 4.

    Holzer, C.: The future of polymer processing. POLIMERI 32(3–4), 124–129 (2011)

    Google Scholar 

  5. 5.

    Kishimoto, A., Buesser, B., Botea, A.: AI meets chemistry. In: The Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18), pp 7978–7982 (2018)

  6. 6.

    Malaguti, E.: The Vertex Coloring Problem and its Generalizations. PhD thesis, Universita Degli Studi di Bologna (2009)

  7. 7.

    Malaguti, E., Toth, P.: A survey on vertex coloring problems. Int. Trans. Oper. Res. 17, 1–34 (2010)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Marx, D.: Graph colouring problems and their applications in scheduling. Period. Polytech. Electr. Eng. 48(1–2), 11–16 (2004)

    Google Scholar 

  9. 9.

    Roy, B.: Nombre chromatique et plus longs chemins d’un graphe. Rev. Fr. Inf. Rech. Opér. (in French) 1, 129–132 (1967)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Schweidtmanna, A.M., Clayton, A.D., Holmes, N., Bradforda, E., Bournec, R.A., Lapkina, A.A.: Machine learning meets continuous flow chemistry: automated optimization towards the Pareto front of multiple objectives. Chem. Eng. J. 352, 277–282 (2018)

    Article  Google Scholar 

  11. 11.

    Spalding, M.A., Chatterjee, A.M.: Handbook of Industrial Polyethylene and Technology: Definitive Guide to Manufacturing, Properties, Processing, Applications and Markets. Wiley, New York (2017)

    Google Scholar 

  12. 12.

    Vitaver, L.M.: Determination of minimal coloring of vertices of a graph by means of Boolean powers of the incidence matrix. Dokl. Akad. Nauk. SSSR 147, 758–759 (in Russian) (1962)

    MathSciNet  MATH  Google Scholar 

Download references


The authors would like to thank the referees for carefully reading our article and for many helpful comments. The first author’s research was supported by JSPS KAKENHI (19K03603).

Author information



Corresponding author

Correspondence to Shinya Fujita.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fujita, S., Kitaev, S., Sato, S. et al. On Properly Ordered Coloring of Vertices in a Vertex-Weighted Graph. Order (2021). https://doi.org/10.1007/s11083-021-09554-7

Download citation


  • Vertex coloring
  • Properly ordered coloring
  • Vertex-weighted graph
  • Gallai-Hasse-Roy-Vitaver theorem