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The number of connected simple cubic graphs on 4, 6, 8, 10, ... vertices is 1, 2, 5, 19, ... (sequence A002851 in the OEIS).A classification according to edge connectivity is made as follows: the 1-connected and 2-connected graphs are defined as usual.
According to Brooks' theorem every connected cubic graph other than the complete graph K 4 has a vertex coloring with at most three colors. Therefore, every connected cubic graph other than K 4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices.
In graph theory, the Möbius ladder M n, for even numbers n, is formed from an n-cycle by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is a cubic, circulant graph, so-named because (with the exception of M 6 (the utility graph K 3,3), M n has exactly n/2 four-cycles [1] which link together by their shared edges to form a topological Möbius strip.
The automorphism group of the Tutte 12-cage is of order 12,096 and is a semi-direct product of the projective special unitary group PSU(3,3) with the cyclic group Z/2Z. [1] It acts transitively on its edges but not on its vertices, making it a semi-symmetric graph, a regular graph that is edge-transitive but not vertex-transitive.
A cubic graph (all vertices have degree three) of girth g that is as small as possible is known as a g-cage (or as a (3,g)-cage).The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage. [3]
When n is congruent to 3 modulo 6 G(n, 2) has exactly three Hamiltonian cycles. [7] For G(n, 2), the number of Hamiltonian cycles can be computed by a formula that depends on the congruence class of n modulo 6 and involves the Fibonacci numbers. [8] Every generalized Petersen graph is a unit distance graph. [9]
The cube-connected cycles of order n is the Cayley graph of a group that acts on binary words of length n by rotation and flipping bits of the word. [1] The generators used to form this Cayley graph from the group are the group elements that act by rotating the word one position left, rotating it one position right, or flipping its first bit.
[27] [28] The skeleton of a cube can be represented as the graph, and it is called the cubical graph, a Platonic graph. It has the same number of vertices and edges as the cube, twelve vertices and eight edges. [29] The cubical graph is a special case of hypercube graph or -cube—denoted as —because it can be constructed by using the ...