## Hamiltonian Graph

Hamiltonian Graph is a connected graph that contains a Hamiltonian Circuit. It means there exists a cycle in the graph that visits each vertex exactly once.

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# Graph

## Hamiltonian Graph

## 5 Color Theorem

## 6 Color Theorem

## Prove every tree is planar

## Prove hypercube is hamiltonian

## Prove that every tree is a bipartite graph

## Show that a graph is bipartite if and only if it does not have a cycle of odd length

## Prove that a bipartite graph has a unique bipartition if and only if it is connected

## Prove that Hypercube is Bipartite

## Prove that Petersen Graph is nonplanar

Hamiltonian Graph is a connected graph that contains a Hamiltonian Circuit. It means there exists a cycle in the graph that visits each vertex exactly once.

According to 5 Color Theorem, every planar graph is 5 colorable. Lemma: Every planar graph is 6 colorable. This is also known as 6 Color Theorem. Proof of 5 Color Theorem We will prove it by mathematical induction. Step 1Every planar graph vertices is 5 colorable. Step 2Assume every planar graph vertices is 5 colorable. …

According to 6 Color Theorem, every planar graph is 6 colorable. Lemma: For any simple planar graph G, the minimum degree is less than equal to 5. Proof: The average degree D of a graph is . We know that every planar graph satisfies the equation . On substituting the equations, we getThus, the average …

Prove that every tree is planar. We can prove every tree is planar by using Euler’s formula or by mathematical induction.

In this article, we will prove that every hypercube is hamiltonian. Hamiltonian: A graph is Hamiltonian if it contains a Hamiltonian circuit. It means there exists a cycle in the graph that visits each vertex exactly once. Hypercube: It is denoted by Qn. It contains 2n vertices and the degree of each vertex is n.Each …

In this article, we will show that every tree is a bipartite graph. Tree: A tree is a simple graph with N – 1 edges where N is the number of vertices such that there is exactly one path between any two vertices. Bipartite: A graph is bipartite if we can divide the vertices into …

In this article, we will prove that a graph is bipartite if and only if it does not have a cycle of odd length. First, we will prove that if a graph is bipartite then it doesn’t have a cycle of odd length. Proof We will prove this by contradiction. Suppose a Bipartite Graph G(V, …

Show that a graph is bipartite if and only if it does not have a cycle of odd length Read More »

In this article, we will show that a bipartite graph has a unique bipartition if and only if it is connected. Bipartite Graph: A Graph is bipartite if we can divide its vertices into two disjoint sets, V1 and V2 such that no edge connects vertices from the same set. Unique Bipartition: Unique Bipartition means …

Prove that a bipartite graph has a unique bipartition if and only if it is connected Read More »

In this article, we will show that every hypercube (Qn) is bipartite. Bipartite: A Graph is Bipartite if we can divide the vertices of the graph into two sets such that no two vertices in the same set are adjacent to each other. Hypercube: A Hypercube is denoted by Qn. A hypercube has 2n vertices …

In this article, we will show that Petersen Graph is non-planar. Petersen Graph: Petersen Graph is a Cubic Graph with 10 vertices and 15 edges such that each vertex has degree 3. There is no 3-cycle or 4-cycle in the Petersen Graph. Non-planar Graph: A graph is called a non-planar graph if it is impossible …