Question

Show that a degree-constrained spanning tree of a simple graph in which each vertex has degree not exceeding 2 consists of a single Hamilton path in the graph.

Short answer

Therefore, the given statement is true. Constrained degree spanning tree of a simple graph in which each vertex has degree not exceeding 2 consists of a single Hamilton path in the graph.

Step-by-step solution

General form

Definition of tree:A tree is a connected undirected graph with no simple circuits.

Definition of Hamilton Path:A simple path in a graph G that passes through every vertex exactly once is called a Hamilton path.

Degree-constrained spanning tree (Definition): A degree-constrained spanning tree of a simple graph G is a spanning tree with the property that the degree of a vertex in this tree cannot exceed some specified bound. Constrained degree spanning trees are useful in models of transportation systems where the number of roads at an intersection is limited, in models of communication networks where the number of connections entering a node is limited, and so on.

Proof of the given statement

Given that, a degree-constrained spanning tree.

Let us take degree-constrained spanning tree as G. And the vertices in G all have a degree of at most 2.

To prove: G consists of a Hamilton path.

As all vertices have a degree of at most 2 and since G is connected, the tree must be a path.

Since this path also passes through every vertex in the tree exactly once which as the tree only contains this path, the path is also a Hamilton path.

Hence it is proved that G consists of a Hamilton path.