Geodesic
More Results on The Smallest One-Realization of A Given Set II
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 473-487.

Voir la notice de l'article provenant de la source Library of Science

Let S be a finite set of positive integers. A mixed hypergraph ℋ is a onerealization of S if its feasible set is S and each entry of its chromatic spectrum is either 0 or 1. The minimum number of vertices, denoted by δ3(S), in a 3-uniform bi-hypergraph which is a one-realization of S was determined in [P. Zhao, K. Diao and F. Lu, More result on the smallest one-realization of a given set, Graphs Combin. 32 (2016) 835–850]. In this paper, we consider the minimum number of edges in a 3-uniform bi-hypergraph which already has the minimum number of vertices with respect of being a minimum bihypergraph that is one-realization of S. A tight lower bound on the number of edges in a 3-uniform bi-hypergraph which is a one-realization of S with δ3(S) vertices is given.
Keywords: mixed hypergraph, feasible set, chromatic spectrum, gap, onerealization 
@article{DMGT_2019_39_2_a11,
     author = {Diao, Kefeng and Lu, Fuliang and Zhao, Ping},
     title = {More {Results} on {The} {Smallest} {One-Realization} of {A} {Given} {Set} {II}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {473--487},
     publisher = {mathdoc},
     volume = {39},
     number = {2},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a11/}
}
TY  - JOUR
AU  - Diao, Kefeng
AU  - Lu, Fuliang
AU  - Zhao, Ping
TI  - More Results on The Smallest One-Realization of A Given Set II
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2019
SP  - 473
EP  - 487
VL  - 39
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a11/
LA  - en
ID  - DMGT_2019_39_2_a11
ER  - 
%0 Journal Article
%A Diao, Kefeng
%A Lu, Fuliang
%A Zhao, Ping
%T More Results on The Smallest One-Realization of A Given Set II
%J Discussiones Mathematicae. Graph Theory
%D 2019
%P 473-487
%V 39
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a11/
%G en
%F DMGT_2019_39_2_a11
Diao, Kefeng; Lu, Fuliang; Zhao, Ping. More Results on The Smallest One-Realization of A Given Set II. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 473-487. http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a11/

[1] G. Bacsó, Zs. Tuza and V. Voloshin, Unique colorings of bi-hypergraphs, Australas. J. Combin. 27 (2003) 33–45.

[2] Cs. Bujtás and Zs. Tuza, Color-bounded hypergraphs, I: General results, Discrete Math. 309 (2009) 4890–4902. doi:10.1016/j.disc.2008.04.019

[3] Cs. Bujtás and Zs. Tuza, Color-bounded hypergraphs, VI: Structural and functional jumps in complexity, Discrete Math. 313 (2013) 1965–1977. doi:10.1016/j.disc.2012.09.020

[4] C. Bujtás and Zs. Tuza, C-perfect hypergraphs, J. Graph Theory 64 (2010) 132–149. doi:10.1002/jgt.20444

[5] Cs. Bujtás and Zs. Tuza, Uniform mixed hypergraphs: the possible numbers of colors, Graphs Combin. 24 (2008) 1–12. doi:10.1007/s00373-007-0765-5

[6] E. Bulgaru and V. Voloshin, Mixed interval hypergraphs, Discrete Appl. Math. 77 (1997) 24–41. doi:10.1016/S0166-218X(97)89209-8

[7] Y. Caro and J. Lauri, Non-monochromatic non-rainbow colourings of σ-hypergraphs, Discrete Math. 318 (2014) 96–104. doi:10.1016/j.disc.2013.11.016

[8] Y. Caro, J. Lauri and C. Zarb, Constrained colouring and σ-hypergraphs, Discuss. Math. Graph Theory 35 (2015) 171–189. doi:10.7151/dmgt.1789

[9] Y. Caro, J. Lauri and C. Zarb, (2, 2) -colourings and clique-free σ-hypergraphs, Discrete Appl. Math. 185 (2015) 38–43. doi:10.1016/j.dam.2014.11.029

[10] K. Diao, G. Liu, D. Rautenbach and P. Zhao, A note on the least number of edges of 3 -uniform hypergraphs with upper chromatic number 2, Discrete Math. 306 (2006) 670–672. doi:10.1016/j.disc.2005.12.020

[11] K. Diao, V. Voloshin, K. Wang and P. Zhao, The smallest one-realization of a given set IV, Discrete Math. 338 (2015) 712–724. doi:10.1016/j.disc.2014.12.021

[12] K. Diao, P. Zhao and K. Wang, The smallest one-realization of a given set III, Graphs Combin. 30 (2014) 875–885. doi:10.1007/s00373-013-1322-z

[13] A. Jaffe, T. Moscibroda and S. Sen, On the price of equivocation in byzantine agreement, in: Proc. 2012 ACM Symposium on Principles of Distributed Computing (ACM, New York, 2012) 309–318. doi:10.1145/2332432.2332491

[14] T. Jiang, D. Mubayi, Zs. Tuza, V. Voloshin and D. West, The chromatic spectrum of mixed hypergraphs, Graphs Combin. 18 (2002) 309–318. doi:10.1007/s003730200023

[15] D. Kobler and A. Kündgen, Gaps in the chromatic spectrum of face-constrained plane graphs, Electron. J. Combin. 8 (2001) #N3.

[16] D. Král, Mixed Hypergraphs and other coloring problems, Discrete Math. 307 (2007) 923–938. doi:10.1016/j.disc.2005.11.050

[17] D. Král, On feasible sets of mixed hypergraphs, Electron. J. Combin. 11 (2004) #R19.

[18] A. Kündgen, E. Mendelsohn and V. Voloshin, Coloring of planar mixed hypergraphs, Electron. J. Combin. 7 (2000) #R60.

[19] V. Voloshin, On the upper chromatic number of a hypergraph, Australas. J. Combin. 11 (1995) 25–45.

[20] V. Voloshin, Coloring Mixed Hypergraphs: Theory, Algorithms and Applications (AMS, Providence, 2002).

[21] V. Voloshin, Mixed Hypergraph Coloring Web Site: http://spectrum.troy.edu/voloshin/mh.html

[22] P. Zhao, K. Diao, R. Chang and K. Wang, The smallest one-realization of a given set II, Discrete Math. 312 (2012) 2946–2951. doi:10.1016/j.disc.2012.06.004

[23] P. Zhao, K. Diao and F. Lu, More result on the smallest one-realization of a given set, Graphs Combin. 32 (2016) 835–850. doi:10.1007/s00373-015-1603-9

[24] P. Zhao, K. Diao and K. Wang, The chromatic spectrum of 3 -uniform bihypergraphs, Discrete Math. 311 (2011) 2650–2656. doi:10.1016/j.disc.2011.08.007

[25] P. Zhao, K. Diao and K. Wang, The smallest one-realization of a given set, Electron. J. Combin. 19 (2012) #P19.