Question

Give an example of each of the following: (i) an infinite bipartite graph; (ii) an infinite connected cubic graph.

Short answer

Part (i): An example of an infinite bipartite graph is an infinite grid. Part (ii): An example of an infinite connected cubic graph is the hexagonal tiling of the plane.

Step-by-step solution

Solving Part (i)

An example of an infinite bipartite graph is an infinite grid, represented as a set of points in a plane with integer coordinates. Such a graph is infinite since there are infinite points, and it is bipartite because the points can be divided into two groups - ones with even sum of coordinates, and ones with odd sum.

Solving Part (ii)

The hexagonal tiling of the plane can be seen as an infinite connected, cubic graph. Each vertex is connected to three others, and you can get from any vertex to any other by a path in the graph. Hence, it is a connected cubic graph.

Key concepts

Bipartite Graph

A bipartite graph is a special type of graph that can be split into two distinct groups of nodes. In a bipartite graph, edges only connect nodes from different groups, never ones within the same group. To visualize this, imagine a group of people separated by interests. Group A likes sports, and Group B likes music. An edge can only exist between someone who likes sports and someone who likes music, but not within Group A or Group B.

A classic example of a bipartite graph is an infinite grid. In this setting, the plane consists of points with integer coordinates. If you assign points with an even sum of coordinates to one group, and points with an odd sum to another, they form a bipartite graph. Since there are infinite points on the grid, this forms an infinite bipartite graph. Additionally, it's worth noting that bipartite graphs are crucial in problems involving matching and network flow. They help us find optimal solutions for pairing or matchmaking scenarios.

Connected Graph

A connected graph is one in which there is a path between every pair of vertices. This means no vertex is isolated; each one is reachable from another through one or more edges. Connected graphs represent many real-life networks, such as social connections or city roadmaps.

For a graph to be connected, you must be able to start from any vertex and find a route to any other. If even one vertex cannot be reached, the graph isn't connected. Infinite graphs can also be connected, although they extend infinitely. The key aspect remains the presence of a pathway that connects any two vertices.

Connected graphs are fundamental in network theory, providing insights into communication networks, transportation systems, and the Internet. Detecting connected components within graphs is essential for analyzing which parts of a system interact.

Cubic Graph

A cubic graph is one where each vertex has exactly three edges, meaning every vertex connects with three others. This classification of graphs has intriguing structures and applications. One famous example is the hexagonal tiling of a plane, known for its neat geometric properties and stability.

When this tiling stretches infinitely, it forms an infinite cubic graph. Each vertex in this graph connects with three others due to the characteristics of hexagons. The graph remains connected since any vertex can reach another by navigating through its paths.

Cubic graphs are essential in understanding complex network systems, especially those requiring stability and uniform distribution of connections, such as electrical grids. By studying cubic graphs, mathematicians and scientists can optimize the efficacy and resilience of diverse network structures.