## What is detour distance?

The detour distance between vertices and of a graph is the length of a longest path between. them, denoted by. In this paper we introduce Detour maximum distance of a graph denoted by. by considering the sum of the degrees of vertices presented in the path, the length of the longest path in addition.

## Is the Petersen graph 3 colorable?

The Petersen graph has chromatic number 3, meaning that its vertices can be colored with three colors — but not with two — such that no edge connects vertices of the same color. It has a list coloring with 3 colors, by Brooks’ theorem for list colorings.

What is the distance in graph theory?

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance.

### Is Petersen graph homeomorphic to K5?

2. Prove that the Petersen graph (shown below) is not planar by finding a subgraph that is homeomorphic to K3,3 or K5. It is impossible to find a subgraph homeomorphic to K5 since we would require a vertex of degree 4.

### Is Herschel graph Hamiltonian?

In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges, the smallest non-Hamiltonian polyhedral graph. It is named after British astronomer Alexander Stewart Herschel.

What is the difference between Euclidean distance and geodesic distance?

The Geodesic distance is the distance of the minimum length inside the figure path and the Euclidean distance is the straight line distance.

#### Why is the Petersen graph not Hamiltonian?

The Petersen graph has no Hamiltonian cycles, but has a Hamiltonian path between any two non-adjacent vertices. In fact, for sufficiently large vertex sets, there is always a graph which admits a Hamiltonian path starting at every vertex, but is not Hamiltonian.

#### How many edges are in K5?

10 edges
K5: K5 has 5 vertices and 10 edges, and thus by Lemma 2 it is not planar. K3,3: K3,3 has 6 vertices and 9 edges, and so we cannot apply Lemma 2.

Who invented Hamiltonian path?

A directed graph in which the path begins and ends on the same vertex (a closed loop) such that each vertex is visited exactly once is known as a Hamiltonian circuit. The 19th-century Irish mathematician William Rowan Hamilton began the systematic mathematical study of such graphs.