A cubic graph is a graph with each vertex having three edges connected to it. A 1-factor is a subgraph of a graph in which each vertex has exactly one edge connected to it. Therefore, a cubic graph containing a 1-factor is a cubic graph with a subgraph in which each vertex has exactly one edge connected to it.
A cubic graph can contain a 1-factor if it has an even number of vertices. This is because each vertex in a 1-factor must have exactly one edge connected to it, and since each edge connects two vertices, the number of edges must be half the number of vertices in order for every vertex to have one edge connected to it.
A cubic graph containing a 1-factor has a special property known as perfect matching. This means that every vertex in the graph is connected to exactly one other vertex, creating a balanced and symmetrical structure. This property is useful in various applications, such as in network design and scheduling problems.
Yes, a cubic graph can contain multiple 1-factors. In fact, the number of 1-factors in a cubic graph can be quite large, depending on the number of vertices. For example, a complete graph with an even number of vertices will have n/2 1-factors, where n is the number of vertices.
A Hamiltonian cycle is a cycle in a graph that visits each vertex exactly once. In a cubic graph, a Hamiltonian cycle would require each vertex to have three edges connected to it, violating the definition of a 1-factor. Therefore, a 1-factor and a Hamiltonian cycle are two different concepts in a cubic graph.