Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001645
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1
{{1,2}}
 => [2] => [1] => ([],1)
 => 1
{{1,2,3}}
 => [3] => [1] => ([],1)
 => 1
{{1,2},{3}}
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
{{1,3},{2}}
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
{{1},{2,3}}
 => [1,2] => [1,1] => ([(0,1)],2)
 => 2
{{1,2,3,4}}
 => [4] => [1] => ([],1)
 => 1
{{1,2,3},{4}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 2
{{1,2,4},{3}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 2
{{1,3,4},{2}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 2
{{1},{2,3,4}}
 => [1,3] => [1,1] => ([(0,1)],2)
 => 2
{{1},{2,3},{4}}
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1},{2,4},{3}}
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1},{2},{3,4}}
 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 4
{{1,2,3,4,5}}
 => [5] => [1] => ([],1)
 => 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 2
{{1,2,3,5},{4}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 2
{{1,2,3},{4,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2
{{1,2,4,5},{3}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 2
{{1,2,4},{3,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2
{{1,2,5},{3,4}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2
{{1,2},{3,4,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 2
{{1,2},{3,4},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 4
{{1,2},{3,5},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 4
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1,3,4,5},{2}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 2
{{1,3,4},{2,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2
{{1,3,5},{2,4}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2
{{1,3},{2,4,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 2
{{1,3},{2,4},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 4
{{1,3},{2,5},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 4
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1,4,5},{2,3}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2
{{1,4},{2,3,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 2
{{1,4},{2,3},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 4
{{1,5},{2,3,4}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 2
{{1},{2,3,4,5}}
 => [1,4] => [1,1] => ([(0,1)],2)
 => 2
{{1},{2,3,4},{5}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1,5},{2,3},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 4
{{1},{2,3,5},{4}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1,4},{2,5},{3}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 4
{{1,4},{2},{3,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1,5},{2,4},{3}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 4
{{1},{2,4,5},{3}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1,5},{2},{3,4}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1},{2},{3,4,5}}
 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 4
{{1,2,3,4,5,6}}
 => [6] => [1] => ([],1)
 => 1
{{1,2,3,4,5},{6}}
 => [5,1] => [1,1] => ([(0,1)],2)
 => 2
{{1,2,3,4,6},{5}}
 => [5,1] => [1,1] => ([(0,1)],2)
 => 2
{{1,2,3,4},{5,6}}
 => [4,2] => [1,1] => ([(0,1)],2)
 => 2
Description
The pebbling number of a connected graph.
Matching statistic: St000454
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 0 = 1 - 1
{{1,2}}
 => [2] => [1] => ([],1)
 => 0 = 1 - 1
{{1,2,3}}
 => [3] => [1] => ([],1)
 => 0 = 1 - 1
{{1,2},{3}}
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,3},{2}}
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1},{2,3}}
 => [1,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,3,4}}
 => [4] => [1] => ([],1)
 => 0 = 1 - 1
{{1,2,3},{4}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,4},{3}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,3,4},{2}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1},{2,3,4}}
 => [1,3] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1},{2,4},{3}}
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1},{2},{3,4}}
 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,3,4,5}}
 => [5] => [1] => ([],1)
 => 0 = 1 - 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,3},{2,4,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,5},{2,3,4}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1},{2,3,4,5}}
 => [1,4] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1},{2,3,4},{5}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1,5},{2,3},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1},{2,3,5},{4}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1,4},{2,5},{3}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,4},{2},{3,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1,5},{2,4},{3}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1},{2,4,5},{3}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1,5},{2},{3,4}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1},{2},{3,4,5}}
 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,3,4,5,6}}
 => [6] => [1] => ([],1)
 => 0 = 1 - 1
{{1,2,3,4,5},{6}}
 => [5,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,3,4,6},{5}}
 => [5,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,3,4},{5,6}}
 => [4,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,3,5,6},{4}}
 => [5,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,3,5},{4,6}}
 => [4,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,3,6},{4,5}}
 => [4,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,3},{4,5},{6}}
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1,2,3},{4,6},{5}}
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1,2,3},{4},{5,6}}
 => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1,2,4,5,6},{3}}
 => [5,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,4,5},{3,6}}
 => [4,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,4,6},{3,5}}
 => [4,2] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1,2,4},{3,5},{6}}
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1},{2},{3,4,5,6}}
 => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,3},{4,5,6},{7}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,3},{4,5,7},{6}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,3},{4,6,7},{5}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,4},{3,5,6},{7}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,4},{3,5,7},{6}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,4},{3,6,7},{5}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,5},{3,4,6},{7}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,5},{3,4,7},{6}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,6},{3,4,5},{7}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,7},{3,4,5},{6}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,6},{3,4,7},{5}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,7},{3,4,6},{5}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2},{3,4},{5,6,7}}
 => [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,5},{3,6,7},{4}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,6},{3,5,7},{4}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2,7},{3,5,6},{4}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2},{3,5},{4,6,7}}
 => [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2},{3,6},{4,5,7}}
 => [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,2},{3,7},{4,5,6}}
 => [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,4},{2,5,6},{7}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,4},{2,5,7},{6}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,4},{2,6,7},{5}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,5},{2,4,6},{7}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,5},{2,4,7},{6}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,6},{2,4,5},{7}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,7},{2,4,5},{6}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,6},{2,4,7},{5}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,7},{2,4,6},{5}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3},{2,4},{5,6,7}}
 => [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,5},{2,6,7},{4}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,6},{2,5,7},{4}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3,7},{2,5,6},{4}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3},{2,5},{4,6,7}}
 => [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3},{2,6},{4,5,7}}
 => [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,3},{2,7},{4,5,6}}
 => [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,4,5},{2,3,6},{7}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,4,5},{2,3,7},{6}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,4,6},{2,3,5},{7}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
{{1,4,7},{2,3,5},{6}}
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001330
(load all 11 compositions to match this statistic)
Mapping
Mp00091: Set partitions —rotate increasing⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%
Values
{{1}}
 => {{1}}
 => [1] => ([],1)
 => 1
{{1,2}}
 => {{1,2}}
 => [2] => ([],2)
 => 1
{{1,2,3}}
 => {{1,2,3}}
 => [3] => ([],3)
 => 1
{{1,2},{3}}
 => {{1},{2,3}}
 => [1,2] => ([(1,2)],3)
 => 2
{{1,3},{2}}
 => {{1,2},{3}}
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => ([],4)
 => 1
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => [1,3] => ([(2,3)],4)
 => 2
{{1,2,4},{3}}
 => {{1,2,3},{4}}
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1},{2,3},{4}}
 => {{1},{2},{3,4}}
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => ([],5)
 => 1
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => [1,4] => ([(3,4)],5)
 => 2
{{1,2,3,5},{4}}
 => {{1,2,3,4},{5}}
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
{{1,2,3},{4,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
{{1,2,4,5},{3}}
 => {{1,2,3,5},{4}}
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
{{1,2,4},{3,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
{{1,2,5},{3,4}}
 => {{1,2,3},{4,5}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
{{1,2},{3,4,5}}
 => {{1,4,5},{2,3}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
{{1,2},{3,5},{4}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
{{1,2},{3},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
{{1,3,4,5},{2}}
 => {{1,2,4,5},{3}}
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
{{1,3,4},{2,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
{{1,3,5},{2,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
{{1,3},{2,4,5}}
 => {{1,3,5},{2,4}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
{{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
{{1,3},{2,5},{4}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
{{1,3},{2},{4,5}}
 => {{1,5},{2,4},{3}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
{{1,4,5},{2,3}}
 => {{1,2,5},{3,4}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
{{1,4},{2,3},{5}}
 => {{1},{2,5},{3,4}}
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
{{1,5},{2,3,4}}
 => {{1,2},{3,4,5}}
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
{{1},{2,3,4,5}}
 => {{1,3,4,5},{2}}
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
{{1},{2,3,4},{5}}
 => {{1},{2},{3,4,5}}
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
{{1,5},{2,3},{4}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
{{1},{2,3,5},{4}}
 => {{1,3,4},{2},{5}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
{{1,4},{2,5},{3}}
 => {{1,3},{2,5},{4}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
{{1,4},{2},{3,5}}
 => {{1,4},{2,5},{3}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
{{1,5},{2,4},{3}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
{{1},{2,4,5},{3}}
 => {{1,3,5},{2},{4}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
{{1,5},{2},{3,4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
{{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
{{1,2,3,4,5,6}}
 => {{1,2,3,4,5,6}}
 => [6] => ([],6)
 => 1
{{1,2,3,4,5},{6}}
 => {{1},{2,3,4,5,6}}
 => [1,5] => ([(4,5)],6)
 => 2
{{1,2,3,4,6},{5}}
 => {{1,2,3,4,5},{6}}
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,3,4},{5,6}}
 => {{1,6},{2,3,4,5}}
 => [2,4] => ([(3,5),(4,5)],6)
 => 2
{{1,2,3,5,6},{4}}
 => {{1,2,3,4,6},{5}}
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,3,5},{4,6}}
 => {{1,5},{2,3,4,6}}
 => [2,4] => ([(3,5),(4,5)],6)
 => 2
{{1,2,3,6},{4,5}}
 => {{1,2,3,4},{5,6}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,3},{4,5},{6}}
 => {{1},{2,3,4},{5,6}}
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2,3},{4,6},{5}}
 => {{1,5},{2,3,4},{6}}
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2,3},{4},{5,6}}
 => {{1,6},{2,3,4},{5}}
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2,4,5,6},{3}}
 => {{1,2,3,5,6},{4}}
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,4,5},{3,6}}
 => {{1,4},{2,3,5,6}}
 => [2,4] => ([(3,5),(4,5)],6)
 => 2
{{1,2,4,6},{3,5}}
 => {{1,2,3,5},{4,6}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,4},{3,5},{6}}
 => {{1},{2,3,5},{4,6}}
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2,4},{3,6},{5}}
 => {{1,4},{2,3,5},{6}}
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2,4},{3},{5,6}}
 => {{1,6},{2,3,5},{4}}
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2,5,6},{3,4}}
 => {{1,2,3,6},{4,5}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,5},{3,4},{6}}
 => {{1},{2,3,6},{4,5}}
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2},{3,4,5,6}}
 => {{1,4,5,6},{2,3}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2},{3,4,5},{6}}
 => {{1},{2,3},{4,5,6}}
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2,6},{3,4},{5}}
 => {{1,2,3},{4,5},{6}}
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2},{3,4,6},{5}}
 => {{1,4,5},{2,3},{6}}
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2,5},{3,6},{4}}
 => {{1,4},{2,3,6},{5}}
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2,5},{3},{4,6}}
 => {{1,5},{2,3,6},{4}}
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2,6},{3,5},{4}}
 => {{1,2,3},{4,6},{5}}
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2},{3,5,6},{4}}
 => {{1,4,6},{2,3},{5}}
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2,6},{3},{4,5}}
 => {{1,2,3},{4},{5,6}}
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2},{3},{4,5,6}}
 => {{1,5,6},{2,3},{4}}
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,2},{3},{4,5},{6}}
 => {{1},{2,3},{4},{5,6}}
 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
{{1,2},{3},{4,6},{5}}
 => {{1,5},{2,3},{4},{6}}
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
{{1,3,4,5,6},{2}}
 => {{1,2,4,5,6},{3}}
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,3,4,5},{2,6}}
 => {{1,3},{2,4,5,6}}
 => [2,4] => ([(3,5),(4,5)],6)
 => 2
{{1,3,4,6},{2,5}}
 => {{1,2,4,5},{3,6}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,3,4},{2,5},{6}}
 => {{1},{2,4,5},{3,6}}
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3,4},{2,6},{5}}
 => {{1,3},{2,4,5},{6}}
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3,4},{2},{5,6}}
 => {{1,6},{2,4,5},{3}}
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3,5,6},{2,4}}
 => {{1,2,4,6},{3,5}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,3,5},{2,4},{6}}
 => {{1},{2,4,6},{3,5}}
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3},{2,4,5,6}}
 => {{1,3,5,6},{2,4}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,3},{2,4,5},{6}}
 => {{1},{2,4},{3,5,6}}
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3,6},{2,4},{5}}
 => {{1,2,4},{3,5},{6}}
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3},{2,4,6},{5}}
 => {{1,3,5},{2,4},{6}}
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3,5},{2,6},{4}}
 => {{1,3},{2,4,6},{5}}
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3,5},{2},{4,6}}
 => {{1,5},{2,4,6},{3}}
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3,6},{2,5},{4}}
 => {{1,2,4},{3,6},{5}}
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3},{2,5,6},{4}}
 => {{1,3,6},{2,4},{5}}
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3,6},{2},{4,5}}
 => {{1,2,4},{3},{5,6}}
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3},{2},{4,5,6}}
 => {{1,5,6},{2,4},{3}}
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,3},{2},{4,5},{6}}
 => {{1},{2,4},{3},{5,6}}
 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
{{1,3},{2},{4,6},{5}}
 => {{1,5},{2,4},{3},{6}}
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
{{1,4,5,6},{2,3}}
 => {{1,2,5,6},{3,4}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,4},{2,3,5,6}}
 => {{1,3,4,6},{2,5}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,5},{2,3,4,6}}
 => {{1,3,4,5},{2,6}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,6},{2,3,4,5}}
 => {{1,2},{3,4,5,6}}
 => [2,4] => ([(3,5),(4,5)],6)
 => 2
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.