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Apollonian network
In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.
An Apollonian network may be formed, starting from a single triangle embedded in the Euclidean plane, by repeatedly selecting a triangular face of the embedding, adding a new vertex inside the face, and connecting the new vertex to each vertex of the face containing it. In this way, the triangle containing the new vertex is subdivided into three smaller triangles, which may in turn be subdivided in the same way.
The complete graphs on three and four vertices, K3 and K4, are both Apollonian networks. K3 is formed by starting with a triangle and not performing any subdivisions, while K4 is formed by making a single subdivision before stopping.
The Goldner–Harary graph is an Apollonian network that forms the smallest non-Hamiltonian maximal planar graph. Another more complicated Apollonian network was used by Nishizeki (1980) to provide an example of a 1-tough non-Hamiltonian maximal planar graph.
As well as being defined by recursive subdivision of triangles, the Apollonian networks have several other equivalent mathematical characterizations. They are the chordal maximal planar graphs, the chordal polyhedral graphs, and the planar 3-trees. They are the uniquely 4-colorable planar graphs, and the planar graphs with a unique Schnyder wood decomposition into three trees. They are the maximal planar graphs with treewidth three, a class of graphs that can be characterized by their forbidden minors or by their reducibility under Y-Δ transforms. They are the maximal planar graphs with degeneracy three. They are also the planar graphs on a given number of vertices that have the largest possible number of triangles, the largest possible number of tetrahedral subgraphs, the largest possible number of cliques, and the largest possible number of pieces after decomposing by separating triangles.
Apollonian networks are examples of maximal planar graphs, graphs to which no additional edges can be added without destroying planarity, or equivalently graphs that can be drawn in the plane so that every face (including the outer face) is a triangle. They are also chordal graphs, graphs in which every cycle of four or more vertices has a diagonal edge connecting two non-consecutive cycle vertices, and the order in which vertices are added in the subdivision process that forms an Apollonian network is an elimination ordering as a chordal graph. This forms an alternative characterization of the Apollonian networks: they are exactly the chordal maximal planar graphs or equivalently the chordal polyhedral graphs.
In an Apollonian network, every maximal clique is a complete graph on four vertices, formed by choosing any vertex and its three earlier neighbors. Every minimal clique separator (a clique that partitions the graph into two disconnected subgraphs) is one of the subdivided triangles. A chordal graph in which all maximal cliques and all minimal clique separators have the same size is a k-tree, and Apollonian networks are examples of 3-trees. Not every 3-tree is planar, but the planar 3-trees are exactly the Apollonian networks.
Every Apollonian network is also a uniquely 4-colorable graph. Because it is a planar graph, the four color theorem implies that it has a graph coloring with only four colors, but once the three colors of the initial triangle are selected, there is only one possible choice for the color of each successive vertex, so up to permutation of the set of colors it has exactly one 4-coloring. It is more difficult to prove, but also true, that every uniquely 4-colorable planar graph is an Apollonian network. Therefore, Apollonian networks may also be characterized as the uniquely 4-colorable planar graphs. Apollonian networks also provide examples of planar graphs having as few k-colorings as possible for k > 4.
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Apollonian network
In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.
An Apollonian network may be formed, starting from a single triangle embedded in the Euclidean plane, by repeatedly selecting a triangular face of the embedding, adding a new vertex inside the face, and connecting the new vertex to each vertex of the face containing it. In this way, the triangle containing the new vertex is subdivided into three smaller triangles, which may in turn be subdivided in the same way.
The complete graphs on three and four vertices, K3 and K4, are both Apollonian networks. K3 is formed by starting with a triangle and not performing any subdivisions, while K4 is formed by making a single subdivision before stopping.
The Goldner–Harary graph is an Apollonian network that forms the smallest non-Hamiltonian maximal planar graph. Another more complicated Apollonian network was used by Nishizeki (1980) to provide an example of a 1-tough non-Hamiltonian maximal planar graph.
As well as being defined by recursive subdivision of triangles, the Apollonian networks have several other equivalent mathematical characterizations. They are the chordal maximal planar graphs, the chordal polyhedral graphs, and the planar 3-trees. They are the uniquely 4-colorable planar graphs, and the planar graphs with a unique Schnyder wood decomposition into three trees. They are the maximal planar graphs with treewidth three, a class of graphs that can be characterized by their forbidden minors or by their reducibility under Y-Δ transforms. They are the maximal planar graphs with degeneracy three. They are also the planar graphs on a given number of vertices that have the largest possible number of triangles, the largest possible number of tetrahedral subgraphs, the largest possible number of cliques, and the largest possible number of pieces after decomposing by separating triangles.
Apollonian networks are examples of maximal planar graphs, graphs to which no additional edges can be added without destroying planarity, or equivalently graphs that can be drawn in the plane so that every face (including the outer face) is a triangle. They are also chordal graphs, graphs in which every cycle of four or more vertices has a diagonal edge connecting two non-consecutive cycle vertices, and the order in which vertices are added in the subdivision process that forms an Apollonian network is an elimination ordering as a chordal graph. This forms an alternative characterization of the Apollonian networks: they are exactly the chordal maximal planar graphs or equivalently the chordal polyhedral graphs.
In an Apollonian network, every maximal clique is a complete graph on four vertices, formed by choosing any vertex and its three earlier neighbors. Every minimal clique separator (a clique that partitions the graph into two disconnected subgraphs) is one of the subdivided triangles. A chordal graph in which all maximal cliques and all minimal clique separators have the same size is a k-tree, and Apollonian networks are examples of 3-trees. Not every 3-tree is planar, but the planar 3-trees are exactly the Apollonian networks.
Every Apollonian network is also a uniquely 4-colorable graph. Because it is a planar graph, the four color theorem implies that it has a graph coloring with only four colors, but once the three colors of the initial triangle are selected, there is only one possible choice for the color of each successive vertex, so up to permutation of the set of colors it has exactly one 4-coloring. It is more difficult to prove, but also true, that every uniquely 4-colorable planar graph is an Apollonian network. Therefore, Apollonian networks may also be characterized as the uniquely 4-colorable planar graphs. Apollonian networks also provide examples of planar graphs having as few k-colorings as possible for k > 4.
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