Abstract
We investigate vector chromatic number (χ vec ), Lovász V-function of the complement \((\bar \vartheta )\), and quantum chromatic number (χ q ) from the perspective of graph homomorphisms. We prove an analog of Sabidussi's theorem for each of these parameters, i.e., that for each of the parameters, the value on the Cartesian product of graphs is equal to the maximum of the values on the factors. Interestingly, as a consequence of this result for \(\bar \vartheta\), we obtain analog of Hedetniemi's conjecture, i.e., that the value of \(\bar \vartheta\) on the categorical product of graphs is equal to the minimum of its values on the factors. We conjecture that the analogous results hold for vector and quantum chromatic number, and we prove that this is the case for some special classes of graphs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Arora and B. Barak: Computational Complexity: A Modern Approach, Cambridge University Press, 2009.
D. Avis, J. Hasegawa, Y. Kikuchi and Y. Sasaki: A quantum protocol to win the graph colouring game on all Hadamard graphs, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E89-A (2006), 1378–1381.
G. Brassard, R. Cleve and A. Tapp: Cost of exactly simulating quantum entan-glement with classical communication, Phys. Rev. Lett. 83 (1999), 1874–1877.
H. Buhrman, R. Cleve and A. Wigderson: Quantum vs. classical communication and computation, in: Proceedings of the thirtieth annual ACM symposium on Theory of computing, STOC ′98, 63–68. ACM, 1998.
P. J. Cameron, A. Montanaro, M. W. Newman, S. Severini and A. Winter: On the quantum chromatic number of a graph, Electr. J. Comb. 14 (2007).
R. Cleve, P. Hoyer, B. Toner and J. Watrous: Consequences and limits of nonlocal strategies, in: 19th IEEE Annual Conference on Computational Complexity, 236–249, 2004.
P. Frankl and Vojtěech Röodl: Forbidden intersections, Trans. Amer. Math. Soc. 300 (1987), 259–286.
J. Fukawa, H. Imai and F. Le Gall: Quantum coloring games via symmetric SAT games, in: Asian Conference on Quantum Information Science (AQIS'11), 2011.
V. Galliard and S. Wolf: Pseudo-telepathy, entanglement, and graph colorings, in: IEEE International Symposium on Information Theory, 101, 2002.
C. Godsil: Equiarboreal graphs, Combinatorica 1 (1981), 163–167.
C. Godsil: Interesting graphs and their colourings, Unpublished notes, 2003.
C. Godsil and G. Royle: Algebraic Graph Theory, Springer-Verlag, New York, 2001.
G. Hahn and C. Tardif: Graph homomorphisms: structure and symmetry, in: Graph symmetry, vol. 497 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 107–166, Kluwer Acad. Publ., 1997.
R. Hammack, W. Imrich and S. Klavžzar: Handbook of Product Graphs, Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL, second edition, 2011.
P. Hell and J. Nešsetřril: Graphs and Homomorphisms, Oxford University Press, 2004.
D. Karger, R. Motwani and M. Sudan: Approximate graph coloring by semidef-inite programming, J. ACM 45 (1998), 246–265.
D. E. Knuth: The sandwich theorem, Electr. J. Comb. 1 (1994).
László Lovász: On the Shannon capacity of a graph, IEEE Trans. Inform. Theory 25 (1979), 1–7.
L. Mančcinska, G. Scarpa and S. Severini: New separations in zero-error channel capacity through projective Kochen-Specker sets and quantum coloring, Advanced online publication, 2013.
R. J. McEliece, E. R. Rodemich and H. C. Rumsey, Jr: The Lovász bound and some generalizations, J. Combin. Inform. System Sci. 3 (1978), 134–152.
M. A. Nielsen and I. L. Chuang: Quantum Computation and Quantum Information, Cambridge University Press, 2000.
D. Roberson and L. Mančcinska: Graph homomorphisms for quantum players, 2012.
G. Sabidussi: Vertex-transitive graphs, Monatsh. Math. 68 (1964), 426–438.
G. Scarpa and S. Severini: Kochen-Specker sets and the rank-1 quantum chromatic number, IEEE Trans. Inf. Theory 58 (2012), 2524–2529.
A. Schrijver: A comparison of the Delsarte and Lovász bounds, IEEE Trans. Inform. Theory 25 (1979), 425–429.
X. Zhu: The fractional version of Hedetniemi's conjecture is true, European J. Combin. 32 (2011), 1168–1175.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Godsil, C., Roberson, D.E., Šámal, R. et al. Sabidussi versus Hedetniemi for three variations of the chromatic number. Combinatorica 36, 395–415 (2016). https://doi.org/10.1007/s00493-014-3132-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-014-3132-1