Values
=>
Cc0020;cc-rep
([],1)=>1
([],2)=>1
([(0,1)],2)=>2
([],3)=>1
([(1,2)],3)=>2
([(0,2),(1,2)],3)=>2
([(0,1),(0,2),(1,2)],3)=>3
([],4)=>1
([(2,3)],4)=>2
([(1,3),(2,3)],4)=>2
([(0,3),(1,3),(2,3)],4)=>2
([(0,3),(1,2)],4)=>2
([(0,3),(1,2),(2,3)],4)=>2
([(1,2),(1,3),(2,3)],4)=>3
([(0,2),(0,3),(1,2),(1,3)],4)=>3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>4
([],5)=>1
([(3,4)],5)=>2
([(2,4),(3,4)],5)=>2
([(1,4),(2,4),(3,4)],5)=>2
([(0,4),(1,4),(2,4),(3,4)],5)=>2
([(1,4),(2,3)],5)=>2
([(1,4),(2,3),(3,4)],5)=>2
([(0,1),(2,4),(3,4)],5)=>2
([(2,3),(2,4),(3,4)],5)=>3
([(0,4),(1,4),(2,3),(3,4)],5)=>2
([(1,3),(1,4),(2,3),(2,4)],5)=>3
([(0,4),(1,3),(2,3),(2,4)],5)=>2
([(0,1),(2,3),(2,4),(3,4)],5)=>3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)=>2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>5
([],6)=>1
([(4,5)],6)=>2
([(3,5),(4,5)],6)=>2
([(2,5),(3,5),(4,5)],6)=>2
([(1,5),(2,5),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>2
([(2,5),(3,4)],6)=>2
([(2,5),(3,4),(4,5)],6)=>2
([(1,2),(3,5),(4,5)],6)=>2
([(3,4),(3,5),(4,5)],6)=>3
([(1,5),(2,5),(3,4),(4,5)],6)=>2
([(0,1),(2,5),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2
([(2,4),(2,5),(3,4),(3,5)],6)=>3
([(0,5),(1,5),(2,4),(3,4)],6)=>2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>2
([(0,5),(1,4),(2,3)],6)=>2
([(1,5),(2,4),(3,4),(3,5)],6)=>2
([(0,1),(2,5),(3,4),(4,5)],6)=>2
([(1,2),(3,4),(3,5),(4,5)],6)=>3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)=>3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)=>3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)=>3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)=>3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)=>2
([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>2
([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>2
([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>2
([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)=>2
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)=>2
([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7)=>2
([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)=>2
([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)=>2
([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)=>2
([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)=>2
([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)=>2
([(0,6),(1,5),(2,4),(3,4),(3,7),(5,7),(6,7)],8)=>2
([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)=>2
([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)=>2
([(0,6),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)=>2
([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)=>2
([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)=>2
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)=>2
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)=>2
([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)=>2
([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)=>2
([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)=>2
([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)=>2
([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)=>2
([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)=>2
([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)=>2
([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)=>2
([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)=>2
([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)=>2
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>8
([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)=>2
([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)=>2
([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)=>2
([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9)=>2
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Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
References
[1] Alon, N., Ben-Eliezer, O., Shangguan, C., Tamo, I. The hat guessing number of graphs arXiv:1812.09752
Code
@cached_function
def get_hat_guessing(N):
d = {}
# complete graphs
for n in range(1,N):
G = graphs.CompleteGraph(n).canonical_label().copy(immutable=True)
d[G] = n
# trees, Butler
for n in range(2,N):
for G in graphs.trees(n):
G = G.canonical_label().copy(immutable=True)
d[G] = 2
# cycles, Szczechla
for n in range(3,N):
G = graphs.CycleGraph(n).canonical_label().copy(immutable=True)
if n == 4 or n % 3 == 0:
d[G] = 3
else:
d[G] = 2
return d
def statistic(G):
"""
Return the hat guessing number, or None.
"""
hat_guessing = get_hat_guessing(10)
HG = 1
for H in G.connected_components_subgraphs():
hg = hat_guessing.get(H.canonical_label().copy(immutable=True), None)
if hg is None:
return
HG = max(HG, hg)
return HG
Created
Dec 27, 2018 at 23:53 by Martin Rubey
Updated
Oct 13, 2019 at 18:44 by Martin Rubey
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