Applications

Some articles about applications of Mixed Hypergraph Coloring  in different areas:

In Ramsey type problems:

• 255. D. Slutzky, V. Voloshin. The Chromatic Spectrum of a Ramsey Mixed Hypergraph. Computer Science Journal of Moldova, vol. 24, no.2(71), 2016, pp. 213-233.
• M. Axenovich, P. Iverson, “Edge-colorings avoiding rainbow and monochromatic subgraphs”, Discrete Math, vol. 308, no. 20, pp.4710–4723, 2008.

In distributed computing (in part related to cyber-security):

• Kinoshita Hirotsugu ; Morizumi Tetsuya. Access Control Model for the Inference Attacks with Access Histories. Computer Software and Applications Conference (COMPSAC), 2017 IEEE 41st Annual.
•  Chuanyou Li; Michel Hurfin; Yun Wang; Lei Yu. Towards a Restrained Use of Non-Equivocation for Achieving Iterative Approximate Byzantine Consensus. 2016 IEEE International Parallel and Distributed Processing Symposium (IPDPS)
•   H. Zhang, L. Song, Z. Han. Radio Resource Allocation for Device-to-Device Underlay Communication Using Hypergraph Theory. IEEE Transactions on Wireless Communications (Volume:15, Issue: 7, 2016).
• S. Sen. New Systems and Algorithms for Scalable Fault Tolerance. PhD Thesis, Princeton University, 2013, 154 p.
• Alexander Jaffe, Thomas Moscibroda, Siddhartha Sen. On the price of equivocation in byzantine agreement. Proceeding PODC ’12 Proceedings of the 2012 ACM symposium on Principles of distributed computing Pages 309-318
  
• Alexander Jaffe, Thomas Moscibroda, Siddhartha Sen. On the Price of Equivocation in Byzantine Agreement. Microsoft Research, Microsoft Corporation 2012.

 

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In WORM and similar/related graph coloring:

/brilliant examples showing that basic concepts of mixed hypergraph coloring provide a wealthy source of infinitely many beautiful problems in graph theory/

• Chandler, James D.; Desormeaux, Wyatt J.; Haynes, Teresa W.; Hedetniemi, Stephen T. Neighborhood-restricted [≤2]-achromatic colorings. Discrete Appl. Math. 207 (2016), 39–44.
•  Honghai Xu. Generalized Colorings of Graphs. PhD Thesis. Clemson University, 2016
•  J. Chandler. Neighborhood-Restricted Achromatic Colorings of Graphs. PhD Thesis. East Tennessee State University, 2016.
•  Cs. Bujtas, Zs. Tuza. F-WORM colorings: Results for 2-connected graphs.
•  Cs. Bujtas, Zs. Tuza. K3-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum. Submitted.
•  MR3308543 Goddard, Wayne; Wash, Kirsti; Xu, Honghai. WORM colorings forbidding cycles or cliques. Congr. Numer. 219 (2014), 161-173.
•  MR3368990 Goddard, Wayne; Wash, Kirsti; Xu, Honghai. Worm colorings. Discuss. Math. Graph Theory 35 (2015), no. 3, 571–584

Special comment.  After you pick a class of graphs and two subgraphs of interest  for C- and D-edges,  the following questions become very interesting immediately:

1. conditions of uncolorability
2. conditions of unique colorability
3. values of the upper and lower chromatic numbers
4. continuity of the chromatic spectrum
5. perfection with respect to the upper chromatic number
6. the values of entries in the chromatic spectrum
7. chromatic polynomial (roots, coefficients)
8. algorithmic aspects  of finding all of the above

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In general graph and hypergraph coloring theory:

• D. Kral. Mixed Hypergraphs and other coloring problems. Discrete Mathematics 307 (7-8) (2007), 923-938.

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In general (range from molecular biology to philosophy):

• V.I. Voloshin. Coloring Mixed Hypergraphs: theory, algorithms and applications. AMS, Providence, 2002.

In coloring of Block Designs:

 

• 228. M. Gionfriddo, V. Voloshin. Bihypergraphs and G-Designs with Broken Chromatic Spectrum: Results and Problems. Applied Mathematical Sciences, Vol. 8, 2014, no. 74, 3673 – 3682.
• 205. J. Matthews. Maximal induced colorable subhypergraphs of all uncolorable BSTS(15)s. Computer Science Journal of Moldova, vol.19, no. 1(55), 2011, pp. 29- 37.
• 185. L. Milazzo, Zs. Tuza. Logarithmic upper bound for the upper chromatic number of $S(t,t+1,v)$ systems. Ars Combin. 92 (2009), 213–223.
• 143. M. Gionfriddo, L. Milazzo, A. Rosa, V. Voloshin. Bicoloring Steiner systems S(2,4,v). Discrete Mathematics, 283 (2004) 249-253.
• 123. L. Gionfriddo. P(3)-designs with gaps in the chromatic spectrum. Rendiconti Seminario Matematico Universita Messina 8 (2002), 49-58.
• 91. M. Buratti, M. Gionfriddo, L. Milazzo, V. Voloshin. Lower and upper chromatic numbers for BSTSs(2^h – 1). Comput. Sci. J. Moldova, Vol. 9, No 2, (2001), 259-272.
• 86. G. Lo Faro, L. Milazzo, A. Tripodi. On the Upper and Lower Chromatic Numbers of BSQSs(16). Electron. J. Combin. 8(1) (2001) R6.
• 83. G. Lo Faro, L. Milazzo, A. Tripodi. The first BSTS with different upper and lower chromatic numbers. Australas. J. Combin. 22 (2000), 123–133.
• 58. Ch. Colbourn, J. Dinitz and A. Rosa. Bicoloring Steiner Triple Systems. Electron. J. Combin. 6 (1999), R25.
• 57. Ch.J. Colbourn, A. Rosa. Triple Systems. Clarendon Press, Oxford, 1999 (section 18.6. Strict colorings and the upper chromatic number, p. 340-341).
• 44. L. Milazzo, Zs. Tuza. Strict Colourings for Classes of Steiner Triple Systems. Discrete Math., 182 (1998) 233-243.

 In Geombinatorics:

• 171. P. Johnson, V. Voloshin. Geombinatorial Mixed Hypergraph Coloring Problems. Geombinatorics, Vol XVII (2), 2007, 57-67.