A Quantum Protocol to Win the Graph Colouring Game on All Hadamard Graphs

David AVIS; Jun HASEGAWA; Yosuke KIKUCHI; Yuuya SASAKI

doi:10.1093/ietfec/e89-a.5.1378

A Quantum Protocol to Win the Graph Colouring Game on All Hadamard Graphs

David AVIS, Jun HASEGAWA, Yosuke KIKUCHI, Yuuya SASAKI

Full Text Views

0

Share
Cite this

Summary :

This paper deals with graph colouring games, an example of pseudo-telepathy, in which two players can convince a verifier that a graph G is c-colourable where c is less than the chromatic number of the graph. They win the game if they convince the verifier. It is known that the players cannot win if they share only classical information, but they can win in some cases by sharing entanglement. The smallest known graph where the players win in the quantum setting, but not in the classical setting, was found by Galliard, Tapp and Wolf and has 32,768 vertices. It is a connected component of the Hadamard graph G_N with N=c=16. Their protocol applies only to Hadamard graphs where N is a power of 2. We propose a protocol that applies to all Hadamard graphs. Combined with a result of Frankl, this shows that the players can win on any induced subgraph of G₁₂ having 1609 vertices, with c=12. Moreover combined with a result of Godsil and Newman, our result shows that all Hadamard graphs G_N (N ≥ 12) and c=N yield pseudo-telepathy games.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E89-A No.5 pp.1378-1381

Publication Date: 2006/05/01

Publicized

Online ISSN: 1745-1337

DOI: 10.1093/ietfec/e89-a.5.1378

Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)

Category

Cite this

Copy

David AVIS, Jun HASEGAWA, Yosuke KIKUCHI, Yuuya SASAKI, "A Quantum Protocol to Win the Graph Colouring Game on All Hadamard Graphs" in IEICE TRANSACTIONS on Fundamentals, vol. E89-A, no. 5, pp. 1378-1381, May 2006, doi: 10.1093/ietfec/e89-a.5.1378.
Abstract: This paper deals with graph colouring games, an example of pseudo-telepathy, in which two players can convince a verifier that a graph G is c-colourable where c is less than the chromatic number of the graph. They win the game if they convince the verifier. It is known that the players cannot win if they share only classical information, but they can win in some cases by sharing entanglement. The smallest known graph where the players win in the quantum setting, but not in the classical setting, was found by Galliard, Tapp and Wolf and has 32,768 vertices. It is a connected component of the Hadamard graph G_N with N=c=16. Their protocol applies only to Hadamard graphs where N is a power of 2. We propose a protocol that applies to all Hadamard graphs. Combined with a result of Frankl, this shows that the players can win on any induced subgraph of G₁₂ having 1609 vertices, with c=12. Moreover combined with a result of Godsil and Newman, our result shows that all Hadamard graphs G_N (N ≥ 12) and c=N yield pseudo-telepathy games.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1093/ietfec/e89-a.5.1378/_p

Copy

@ARTICLE{e89-a_5_1378,
author={David AVIS, Jun HASEGAWA, Yosuke KIKUCHI, Yuuya SASAKI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Quantum Protocol to Win the Graph Colouring Game on All Hadamard Graphs},
year={2006},
volume={E89-A},
number={5},
pages={1378-1381},
abstract={This paper deals with graph colouring games, an example of pseudo-telepathy, in which two players can convince a verifier that a graph G is c-colourable where c is less than the chromatic number of the graph. They win the game if they convince the verifier. It is known that the players cannot win if they share only classical information, but they can win in some cases by sharing entanglement. The smallest known graph where the players win in the quantum setting, but not in the classical setting, was found by Galliard, Tapp and Wolf and has 32,768 vertices. It is a connected component of the Hadamard graph G_N with N=c=16. Their protocol applies only to Hadamard graphs where N is a power of 2. We propose a protocol that applies to all Hadamard graphs. Combined with a result of Frankl, this shows that the players can win on any induced subgraph of G₁₂ having 1609 vertices, with c=12. Moreover combined with a result of Godsil and Newman, our result shows that all Hadamard graphs G_N (N ≥ 12) and c=N yield pseudo-telepathy games.},
keywords={},
doi={10.1093/ietfec/e89-a.5.1378},
ISSN={1745-1337},
month={May},}

Copy

TY - JOUR
TI - A Quantum Protocol to Win the Graph Colouring Game on All Hadamard Graphs
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1378
EP - 1381
AU - David AVIS
AU - Jun HASEGAWA
AU - Yosuke KIKUCHI
AU - Yuuya SASAKI
PY - 2006
DO - 10.1093/ietfec/e89-a.5.1378
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E89-A
IS - 5
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - May 2006
AB - This paper deals with graph colouring games, an example of pseudo-telepathy, in which two players can convince a verifier that a graph G is c-colourable where c is less than the chromatic number of the graph. They win the game if they convince the verifier. It is known that the players cannot win if they share only classical information, but they can win in some cases by sharing entanglement. The smallest known graph where the players win in the quantum setting, but not in the classical setting, was found by Galliard, Tapp and Wolf and has 32,768 vertices. It is a connected component of the Hadamard graph G_N with N=c=16. Their protocol applies only to Hadamard graphs where N is a power of 2. We propose a protocol that applies to all Hadamard graphs. Combined with a result of Frankl, this shows that the players can win on any induced subgraph of G₁₂ having 1609 vertices, with c=12. Moreover combined with a result of Godsil and Newman, our result shows that all Hadamard graphs G_N (N ≥ 12) and c=N yield pseudo-telepathy games.
ER -