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How do you determine if a graph has a triangle?

How do you determine if a graph has a triangle?

We define a triangle in a graph as any triple of vertices {u, v, w} so that there is an edge between any two of them in G.

How do you prove a graph is triangle free?

A graph is said to be triangle free if no two adjacent vertices are adjacent to a common vertex.

How do you show that a graph is complete?

To be a complete graph:

  1. The number of edges in the graph must be N(N-1)/2.
  2. Each vertice must be connected to exactly N-1 other vertices.

How do you prove that a graph has a cycle?

Proof: Let G be a graph with n vertices. If G is connected then by theorem 3 it is not a tree, so it contains a cycle. If G is not connected, one of its connected components has at least as many edges as vertices so this component is not a tree and must contain a cycle, hence G contains a cycle.

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What is a triangle in a graph?

A triangle is a set of 3 vertices such that any two of them are connected by an edge of the graph; the graph is then said to “contain” that triangle. Theorem Let M.V be the maximum number of edges of a graph with V vertices such that the graph need not contain a triangle; then M .

Is Grotzsch graph Hamiltonian?

The Grötzsch graph is smallest triangle-free graph with chromatic number four. It is identical to the Mycielski graph of order four, and is implemented as GraphData[“GrotztschGraph”]. It has 11 vertices and 20 edges. It is Hamiltonian, but nonplanar.

How do you prove that graph is connected?

A graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices. An undirected graph that is not connected is called disconnected.

How do you print a cycle on a graph?

Print all the cycles in an undirected graph

  1. Insert the edges into an adjacency list.
  2. Call the DFS function which uses the coloring method to mark the vertex.
  3. Whenever there is a partially visited vertex, backtrack till the current vertex is reached and mark all of them with cycle numbers.
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Is a triangle a complete graph?

In the mathematical field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle….

Triangle graph
Chromatic number 3
Chromatic index 3
Properties 2-regular Vertex-transitive Edge-transitive Unit distance Hamiltonian Eulerian
Notation or