Graph Theory - Girth and Circumference

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Girth of a Graph

The girth of a graph is defined as the length of the shortest cycle within the graph. If the graph does not contain any cycles (i.e., it is acyclic), its girth is considered to be infinite.

Girth provides details of the smallest cyclic structure within the graph, which is useful in various graph theory applications and analyses.

Calculating the Girth

To compute the girth of a graph, follow these steps −

• Identify all cycles in the graph.
• Determine the length of each cycle.
• The girth is the length of the shortest cycle.

Example: Simple Graph Girth

Consider the following graph with 6 vertices −

Let us identify the cycles in the graph −

• Cycle 1: 1 - 2 - 3 - 1 (length = 3)
• Cycle 2: 2 - 3 - 4 - 2 (length = 3)
• Cycle 3: 3 - 4 - 5 - 3 (length = 3)
• Cycle 4: 2 - 3 - 5 - 2 (length = 3)
• Cycle 5: 2 - 3 - 5 - 4 - 2 (length = 4)

Therefore, the girth of the graph is 3, as the shortest cycles have a length of 3.

Importance of Girth

Understanding the girth of a graph is important for various reasons −

• It provides information about the smallest cycle in the graph, which is crucial for cycle detection algorithms.
• Graphs with large girths are sparse and have fewer short cycles, which is important in network design and analysis.
• Girth is used in studying extremal graph theory and properties of random graphs.

Girth in Tree

A tree is a connected graph with no cycles. Therefore, the girth of any tree is infinite, as there are no cycles present. Trees have the property that there is exactly one path between any pair of vertices, making them minimally connected. A tree with n vertices has exactly n1 edges.

For example, consider the following tree graph −

Since the graph has no cycles, its girth is infinite.

Girth of Complete Graphs

A complete graph K_n is a graph where there is an edge between every pair of vertices. The girth of a complete graph with more than 2 vertices is 3, as any three vertices form a triangle, which is the shortest cycle.

Girth of Bipartite Graphs

A bipartite graph is a graph whose vertices can be divided into two disjoint sets such that no two graph vertices within the same set are adjacent.

If a bipartite graph contains cycles, all such cycles must be of even length. The girth of a bipartite graph is the length of its shortest cycle, which will be an even number.

Circumference of a Graph

The circumference of a graph is defined as the length of the longest cycle within the graph. If the graph does not contain any cycles, its circumference is considered to be zero.

Circumference helps in identifying the largest cyclic structure within the graph, providing details into the overall structure and complexity of the graph.

Calculating the Circumference

To compute the circumference of a graph, follow these steps −

• Identify all cycles in the graph.
• Determine the length of each cycle.
• The circumference is the length of the longest cycle.

Example: Simple Graph Circumference

Consider the same graph used in the girth example. We previously identified the cycles in the graph −

• Cycle 1: 1 - 2 - 3 - 1 (length = 3)
• Cycle 2: 2 - 3 - 4 - 2 (length = 3)
• Cycle 3: 3 - 4 - 5 - 3 (length = 3)
• Cycle 4: 2 - 3 - 5 - 2 (length = 3)
• Cycle 5: 2 - 3 - 5 - 4 - 2 (length = 4)

Therefore, the circumference of the graph is 4, as the longest cycle has a length of 4.

Importance of Circumference

Understanding the circumference of a graph is important for several reasons, they are −

• It provides information about the largest cycle in the graph, which is crucial for analyzing the graph's complexity.
• Graphs with large circumferences are complex and have long cyclic structures, which is important in network analysis and design.
• Circumference is used in studying the properties of random graphs and extremal graph theory.

Circumference of Tree

As trees do not contain any cycles, their circumference is not defined. Trees are acyclic by definition, which is why their girth is infinite and circumference is considered nonexistent (zero).

Circumference of Complete Graphs

In a complete graph K_n, the longest cycle includes all n vertices. Therefore, the circumference of a complete graph K_n is n, as every vertex can be included in the cycle.

Circumference of Cyclic Graphs

In a cycle graph C_n (a single cycle with n vertices), the circumference is n, as this is the only cycle present. For more complex cyclic graphs, the circumference is the length of the longest cycle found within the graph.

In this graph, the circumference is 3.

Girth and Circumference of Various Graphs

Here are some examples of calculating the girth and circumference in different types of graphs −

Graph Type	Girth	Circumference
Path Graph	Infinite (no cycles)	0 (no cycles)
Star Graph	Infinite (no cycles)	0 (no cycles)
Wheel Graph	3 (smallest cycle)	n (cycle through all vertices)
Grid Graph	4 (smallest square cycle)	Depends on grid dimensions