1. bookVolume 39 (2019): Issue 2 (May 2019)
Journal Details
License
Format
Journal
First Published
13 Apr 2013
Publication timeframe
4 times per year
Languages
English
Copyright
© 2020 Sciendo

More Results on The Smallest One-Realization of A Given Set II

Published Online: 26 Jan 2019
Page range: 473 - 487
Received: 22 Jun 2016
Accepted: 15 Sep 2017
Journal Details
License
Format
Journal
First Published
13 Apr 2013
Publication timeframe
4 times per year
Languages
English
Copyright
© 2020 Sciendo

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

MSC 2010

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

[2] Cs. Bujtás and Zs. Tuza, Color-bounded hypergraphs, I: General results, Discrete Math. 309 (2009) 4890–4902. doi:10.1016/j.disc.2008.04.01910.1016/j.disc.2008.04.019Open DOISearch in Google Scholar

[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.2012.09.02010.1016/j.disc.2012.09.020Open DOISearch in Google Scholar

[4] C. Bujtás and Zs. Tuza, C-perfect hypergraphs, J. Graph Theory 64 (2010) 132–149. doi:10.1002/jgt.2044410.1002/jgt.20444Open DOISearch in Google Scholar

[5] Cs. Bujtás and Zs. Tuza, Uniform mixed hypergraphs: the possible numbers of colors, Graphs Combin. 24 (2008) 1–12. doi:10.1007/s00373-007-0765-510.1007/s00373-007-0765-5Open DOISearch in Google Scholar

[6] E. Bulgaru and V. Voloshin, Mixed interval hypergraphs, Discrete Appl. Math. 77 (1997) 24–41. doi:10.1016/S0166-218X(97)89209-810.1016/S0166-218X(97)89209-8Open DOISearch in Google Scholar

[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.2013.11.01610.1016/j.disc.2013.11.016Open DOISearch in Google Scholar

[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.1789Search in Google Scholar

[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.2014.11.02910.1016/j.dam.2014.11.029Open DOISearch in Google Scholar

[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.2005.12.020Search in Google Scholar

[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.2014.12.02110.1016/j.disc.2014.12.021Open DOISearch in Google Scholar

[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-z10.1007/s00373-013-1322-zOpen DOISearch in Google Scholar

[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:10.1145/2332432.233249110.1145/2332432.2332491Open DOISearch in Google Scholar

[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/s00373020002310.1007/s003730200023Open DOISearch in Google Scholar

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

[16] D. Král, Mixed Hypergraphs and other coloring problems, Discrete Math. 307 (2007) 923–938. doi:10.1016/j.disc.2005.11.05010.1016/j.disc.2005.11.050Open DOISearch in Google Scholar

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

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

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

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

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

[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.2012.06.00410.1016/j.disc.2012.06.004Open DOISearch in Google Scholar

[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/s00373-015-1603-910.1007/s00373-015-1603-9Open DOISearch in Google Scholar

[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.2011.08.00710.1016/j.disc.2011.08.007Open DOISearch in Google Scholar

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

Plan your remote conference with Sciendo