# Cubic graph containing a 1-factor

• Robb
In summary, the conversation discusses how to show the existence of a 1-factor in a given graph and the concept of bridges. A 1-factor is the same as a perfect matching and a graph has a perfect matching if a certain condition is met. It is also mentioned that every bridgeless cubic graph and every cubic graph with at most two bridges contains a 1-factor. The conversation also discusses the number of vertices in each component of a connected graph after removing the bridges. A solution is provided using pentagonal circuits and a triangle.
Robb
Homework Statement
Does there exist a cubic graph with three bridges that contains a 1-factor?
Relevant Equations
##k_0 (G-S) \leq |S|##

##k_0 (G-S) \leq |S|
I understand how to show a given graph does/does not contain a 1-factor but I'm not sure how to show existence (or the lack thereof). Please advise.

Please define 'factor' here. I am familiar with the concept of Bridges but not with the concept of factors. And maybe you can add tags at the end of your expression so the Latex can render?

1-factor is the same as a perfect matching-A graph G has a perfect matching iff for ##S \in V(G), k_o(G-S) \leq |S|##. If G has odd order, then G has no 1-factor. A 1-factor is a 1-regular sub-graph of G.
Also, every bridgeless cubic graph contains a 1-factor as well as every cubic graph with at most two bridges contains a 1-factor.

WWGD
Restrict attention to a connected graph. If you remove the bridges the graph will fall into a number of components. What can you say about the number of vertices in each?

Since the thread seems to have died, here's my solution.

Connect each vertex of the triangle to a vertex of a pentagon to make the graph connected with three bridges.
Within each pentagon, connect the remaining four vertices as two pairs, making the graph cubic.

## 1. What is a cubic graph containing a 1-factor?

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.

## 2. How can a cubic graph contain a 1-factor?

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.

## 3. What is the significance of a cubic graph containing a 1-factor?

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.

## 4. Can a cubic graph contain more than one 1-factor?

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.

## 5. How is a 1-factor different from a Hamiltonian cycle in a cubic graph?

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.

Replies
4
Views
1K
Replies
7
Views
960
Replies
3
Views
2K
Replies
1
Views
964
Replies
6
Views
1K
Replies
2
Views
1K
Replies
9
Views
1K
Replies
6
Views
946
Replies
8
Views
2K
Replies
32
Views
2K