Values
([],1) => ([],1) => ([(0,1)],2) => 2
([],2) => ([],1) => ([(0,1)],2) => 2
([(0,1)],2) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([],3) => ([],1) => ([(0,1)],2) => 2
([(1,2)],3) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,2),(1,2)],3) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([],4) => ([],1) => ([(0,1)],2) => 2
([(2,3)],4) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,3),(2,3)],4) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(1,2)],4) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
([],5) => ([],1) => ([(0,1)],2) => 2
([(3,4)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,4),(2,3)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,1),(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],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
([],6) => ([],1) => ([(0,1)],2) => 2
([(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(2,5),(3,4)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,2),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,4),(2,3)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
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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.
Map
cone
Description
The cone of a graph.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.
Map
core
Description
The core of a graph.
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
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