Graphs and Combinatorics

, Volume 33, Issue 4, pp 869–883 | Cite as

The Smallest Uniform Color-Bounded Hypergraphs Which are One-Realizations of a Given Set

  • Kefeng Diao
  • Fuliang Lu
  • Vitaly Voloshin
  • Ping ZhaoEmail author
Original Paper


A color-bounded hypergraph is a hypergraph with vertex set X and edge set \({\mathcal {E}}=\{E_1,E_2,\dots ,E_m\}\), together with integers \(s_i\) and \(t_i\) (\(1\le s_i\le t_i\le |E_i|\)) for \(i=1,2,\ldots ,m\). A vertex coloring \(\varphi \) is proper if the number of colors occurring in edge \(E_i\) satisfies \(s_i\le |\varphi (E_i)|\le t_i\), for every \(1\le i\le m\). If \(s_i=s\) and \(t_i=t\) for all i, we simply denote the color-bounded hypergraph by \({\mathcal {H}}=(X, {\mathcal {E}},s,t)\). A set of positive integers \(\Phi (\mathcal {H})\) is called feasible, if it consists of all k for which there exists a proper coloring of \(\mathcal {H}\) using precisely k colors. Chromatic spectrum of a hypergraph \(\mathcal {H}\) is a vector with each entry \(r_k\) equal to the number of partitions of vertex set induced by all proper colorings using k colors. Let S be a finite set of positive integers. A color-bounded hypergraph is a one-realization of S if its feasible set is S and each entry of its chromatic spectrum is either 0 or 1. In this paper, we determine the minimum number of vertices of r-uniform color-bounded hypergraphs \({\mathcal {H}}=(X, {\mathcal {E}},2,t)\) which are one-realizations of S for the case when \(\lceil \frac{r}{2}\rceil <t\le r-2\) and \(\max (S)\ge \frac{3r}{2}\).


Hypergraph coloring Color-bounded hypergraph Feasible set Chromatic spectrum Gap One-realization 



We thanks the referees for helpful suggestions.


  1. 1.
    Axenovich, M., Kündgen, A.: On a generalized anti-Ramsey problem. Combinatorica 21, 335–349 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bujtás, C., Tuza, Z.: Color-bounded hypergraphs, I: general results. Discret. Math. 309, 4890–4902 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bujtás, C., Tuza, Z.: Color-bounded hypergraphs, II: interval hypergraphs and hypertrees. Discret. Math. 309, 6391–6401 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bujtás, C., Tuza, Z.: Color-bounded hypergraphs, III: model comparison. Appl. Anal. Discret. Math. 1, 36–55 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bujtás, C., Tuza, Z.: Color-bounded hypergraphs, IV: stable colorings of hypertrees. Discret. Math. 310, 1463–1474 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bujtás, C., Tuza, Z., Voloshin, V.: Color-bounded hypergraphs, V: host graphs and subdivisions. Discuss. Math. Graph Theory 31(2), 223–238 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bulgaru, E., Voloshin, V.: Mixed interval hypergraphs. Discret. Appl. Math. 77, 24–41 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Diao, K., Liu, G., Rautenbach, D., Zhao, P.: A note on the least number of edges of 3-uniform hypergraphs with upper chromatic number 2. Discret. Math. 306, 670–672 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Diao, K., Voloshin, V., Wang, K., Zhao, P.: The smallest one-realization of a given set IV. Discret. Math. 338, 712–724 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Diao, K., Zhao, P., Wang, K.: The smallest one-realization of a given set III. Graphs Comb. 30, 875–885 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Diao, K., Zhao, P., Zhou, H.: About the upper chromatic number of a co-hypergraph. Discret. Math. 220, 249–259 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Drgas-Burchardt, E., Łazuka, E.: On chromatic polynomials of hypergraphs. Appl. Math. Lett. 20(12), 1250–1254 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dvořák, D., Kára, J., Král, D., Pangrác, O.: Pattern hypergraphs. Electron. J. Combin 17, R15 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jaffe, A., Moscibroda, T., Sen, S.: On the price of equivocation in byzantine agreement. In: Proc. 31st Principles of Distributed Computing (PODC) (2012)Google Scholar
  15. 15.
    Jiang, T.: Edge-colorings with no large polychromatic stars. Graphs Comb. 18, 303–308 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jiang, T., Mubayi, D., Tuza, Zs, Voloshin, V., West, D.: The chromatic spectrum of mixed hypergraphs. Graphs Comb. 18, 309–318 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jiang, T., West, D.: Edge-colorings of complete graphs that avoid polychromatic trees. Discret. Math. 274, 137–145 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jiang, T., West, D.: On the Erdős-Simonovits-Sós conjecture about the anti-Ramsey number of a cycle. Comb. Probab. Comput. 12, 585–598 (2003)CrossRefzbMATHGoogle Scholar
  19. 19.
    Kobler, D., Kündgen, A.: Gaps in the chromatic spectrum of faced-constrained palne graphs. Electron. J. Comb. 3, N3 (2001)zbMATHGoogle Scholar
  20. 20.
    Král, D.: A counter-example to Voloshin’s hypergraphs co-perfectness conjecture. Australas. J. Comb. 27, 253–262 (2003)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Král, D.: On feasible sets of mixed hypergraphs. Electron. J. Comb. 11, R19 (2004)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Král, D.: Mixed hypergraphs and other coloring problems. Discret. Math. 307(7–8), 923–938 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kündgen, A., Mendelsohn, E., Voloshin, V.: Coloring of planar mixed hypergraphs. Electron. J. Comb. 7, R60 (2000)zbMATHGoogle Scholar
  24. 24.
    Voloshin, V.: On the upper chromatic number of a hypergraph. Australas. J. Comb. 11, 25–45 (1995)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Voloshin, V.: Coloring Mixed Hypergraphs: Theory, Algorithms and Applications. AMS, Providence (2002)zbMATHGoogle Scholar
  26. 26.
    Voloshin, V.: Mixed Hypergraph Coloring Web Site:
  27. 27.
    Zhao, P., Diao, K., Chang, R., Wang, K.: The smallest one-realization of a given set II. Discret. Math. 312, 2946–2951 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zhao, P., Diao, K., Wang, K.: The chromatic spectrum of 3-uniform bi-hypergraphs. Discret. Math. 311, 2650–2656 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zhao, P., Diao, K., Wang, K.: The smallest one-realization of a given set. Electron. J. Combin 19, P19 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Japan 2017

Authors and Affiliations

  • Kefeng Diao
    • 1
  • Fuliang Lu
    • 1
  • Vitaly Voloshin
    • 2
  • Ping Zhao
    • 1
    Email author
  1. 1.School of ScienceLinyi UniversityLinyiChina
  2. 2.Department of MathematicsTroy UniversityTroyUSA

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