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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 a cubic graph with a perfect matching, the edges that are not in the perfect matching form a 2-factor. By orienting the 2-factor, the edges of the perfect matching can be extended to paths of length three, say by taking the outward-oriented edges. This shows that every cubic, bridgeless graph decomposes into edge-disjoint paths of length ...
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]
every cubic bridgeless graph without Petersen minor has a cycle double cover. [13] is the smallest cubic graph with Colin de Verdière graph invariant μ = 5. [14] is the smallest graph of cop number 3. [15] has radius 2 and diameter 2. It is the largest cubic graph with diameter 2. [b] has 2000 spanning trees, the most of any 10-vertex cubic ...
The Möbius–Kantor graph is a subgraph of the four-dimensional hypercube graph, formed by removing eight edges from the hypercube. [1] Since the hypercube is a unit distance graph, the Möbius–Kantor graph can also be drawn in the plane with all edges unit length, although such a drawing will necessarily have some pairs of crossing edges.
The cube-connected cycles of order n (denoted CCC n) can be defined as a graph formed from a set of n2 n nodes, indexed by pairs of numbers (x, y) where 0 ≤ x < 2 n and 0 ≤ y < n. Each such node is connected to three neighbors: ( x , ( y + 1) mod n ) , ( x , ( y − 1) mod n ) , and ( x ⊕ 2 y , y ) , where "⊕" denotes the bitwise ...
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]