Identifier
Mp00184:
Integer compositions
—to threshold graph⟶
Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Images
[1] => ([],1) => [1] => [1]
[1,1] => ([(0,1)],2) => [1,1] => [2]
[2] => ([],2) => [2] => [1,1]
[1,1,1] => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3]
[1,2] => ([(1,2)],3) => [2,1] => [2,1]
[2,1] => ([(0,2),(1,2)],3) => [2,1] => [2,1]
[3] => ([],3) => [3] => [1,1,1]
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4]
[1,1,2] => ([(1,2),(1,3),(2,3)],4) => [2,1,1] => [3,1]
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => [2,1,1] => [3,1]
[1,3] => ([(2,3)],4) => [3,1] => [2,1,1]
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [2,1,1] => [3,1]
[2,2] => ([(1,3),(2,3)],4) => [3,1] => [2,1,1]
[3,1] => ([(0,3),(1,3),(2,3)],4) => [3,1] => [2,1,1]
[4] => ([],4) => [4] => [1,1,1,1]
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [5]
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [4,1]
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [4,1]
[1,1,3] => ([(2,3),(2,4),(3,4)],5) => [3,1,1] => [3,1,1]
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [4,1]
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [3,1,1]
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [3,1,1]
[1,4] => ([(3,4)],5) => [4,1] => [2,1,1,1]
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [4,1]
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [3,1,1]
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [3,1,1]
[2,3] => ([(2,4),(3,4)],5) => [4,1] => [2,1,1,1]
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [3,1,1]
[3,2] => ([(1,4),(2,4),(3,4)],5) => [4,1] => [2,1,1,1]
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => [4,1] => [2,1,1,1]
[5] => ([],5) => [5] => [1,1,1,1,1]
[1,1,1,1,1,1] => ([(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) => [1,1,1,1,1,1] => [6]
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [2,1,1,1,1] => [5,1]
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [2,1,1,1,1] => [5,1]
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [4,1,1]
[1,1,2,1,1] => ([(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) => [2,1,1,1,1] => [5,1]
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [4,1,1]
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [4,1,1]
[1,1,4] => ([(3,4),(3,5),(4,5)],6) => [4,1,1] => [3,1,1,1]
[1,2,1,1,1] => ([(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) => [2,1,1,1,1] => [5,1]
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [4,1,1]
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [4,1,1]
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [3,1,1,1]
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [4,1,1]
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [3,1,1,1]
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [3,1,1,1]
[1,5] => ([(4,5)],6) => [5,1] => [2,1,1,1,1]
[2,1,1,1,1] => ([(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) => [2,1,1,1,1] => [5,1]
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [4,1,1]
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [4,1,1]
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [3,1,1,1]
[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) => [3,1,1,1] => [4,1,1]
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [3,1,1,1]
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [3,1,1,1]
[2,4] => ([(3,5),(4,5)],6) => [5,1] => [2,1,1,1,1]
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [4,1,1]
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [3,1,1,1]
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [3,1,1,1]
[3,3] => ([(2,5),(3,5),(4,5)],6) => [5,1] => [2,1,1,1,1]
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [3,1,1,1]
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => [5,1] => [2,1,1,1,1]
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5,1] => [2,1,1,1,1]
[6] => ([],6) => [6] => [1,1,1,1,1,1]
[1,1,1,1,1,1,1] => ([(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) => [1,1,1,1,1,1,1] => [7]
[1,1,1,1,1,2] => ([(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) => [2,1,1,1,1,1] => [6,1]
[1,1,1,1,2,1] => ([(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) => [2,1,1,1,1,1] => [6,1]
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [5,1,1]
[1,1,1,2,1,1] => ([(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) => [2,1,1,1,1,1] => [6,1]
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [5,1,1]
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [5,1,1]
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
[1,1,2,1,1,1] => ([(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) => [2,1,1,1,1,1] => [6,1]
[1,1,2,1,2] => ([(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) => [3,1,1,1,1] => [5,1,1]
[1,1,2,2,1] => ([(0,6),(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) => [3,1,1,1,1] => [5,1,1]
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
[1,1,3,1,1] => ([(0,5),(0,6),(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) => [3,1,1,1,1] => [5,1,1]
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
[1,1,5] => ([(4,5),(4,6),(5,6)],7) => [5,1,1] => [3,1,1,1,1]
[1,2,1,1,1,1] => ([(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) => [2,1,1,1,1,1] => [6,1]
[1,2,1,1,2] => ([(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) => [3,1,1,1,1] => [5,1,1]
[1,2,1,2,1] => ([(0,6),(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) => [3,1,1,1,1] => [5,1,1]
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
[1,2,2,1,1] => ([(0,5),(0,6),(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) => [3,1,1,1,1] => [5,1,1]
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [3,1,1,1,1]
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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) => [3,1,1,1,1] => [5,1,1]
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [3,1,1,1,1]
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [3,1,1,1,1]
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [3,1,1,1,1]
[1,6] => ([(5,6)],7) => [6,1] => [2,1,1,1,1,1]
[2,1,1,1,1,1] => ([(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) => [2,1,1,1,1,1] => [6,1]
[2,1,1,1,2] => ([(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) => [3,1,1,1,1] => [5,1,1]
[2,1,1,2,1] => ([(0,6),(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) => [3,1,1,1,1] => [5,1,1]
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
[2,1,2,1,1] => ([(0,5),(0,6),(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) => [3,1,1,1,1] => [5,1,1]
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [4,1,1,1]
>>> Load all 127 entries. <<<Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
Map
chromatic difference sequence
Description
The chromatic difference sequence of a graph.
Let $G$ be a simple graph with chromatic number $\kappa$. Let $\alpha_m$ be the maximum number of vertices in a $m$-colorable subgraph of $G$. Set $\delta_m=\alpha_m-\alpha_{m-1}$. The sequence $\delta_1,\delta_2,\dots\delta_\kappa$ is the chromatic difference sequence of $G$.
All entries of the chromatic difference sequence are positive: $\alpha_m > \alpha_{m-1}$ for $m < \kappa$, because we can assign any uncolored vertex of a partial coloring with $m-1$ colors the color $m$. Therefore, the chromatic difference sequence is a composition of the number of vertices of $G$ into $\kappa$ parts.
Let $G$ be a simple graph with chromatic number $\kappa$. Let $\alpha_m$ be the maximum number of vertices in a $m$-colorable subgraph of $G$. Set $\delta_m=\alpha_m-\alpha_{m-1}$. The sequence $\delta_1,\delta_2,\dots\delta_\kappa$ is the chromatic difference sequence of $G$.
All entries of the chromatic difference sequence are positive: $\alpha_m > \alpha_{m-1}$ for $m < \kappa$, because we can assign any uncolored vertex of a partial coloring with $m-1$ colors the color $m$. Therefore, the chromatic difference sequence is a composition of the number of vertices of $G$ into $\kappa$ parts.
Map
conjugate
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