What Does a Graph with Three Triangle Components Have?

The description “a graph with 3 components which are triangles” specifies a graph’s structure but not which property you want to find. Under the standard interpretation, it is the disjoint union of three copies of the complete graph $K_3$. That lets you determine many basic properties directly.

A triangle here means a graph with three vertices and all three possible edges between them. It does not necessarily mean that the graph is drawn as a geometric equilateral triangle. The key point is that the three copies have no vertices or edges in common.

Write the graph as a disjoint union

The graph can be denoted by

$$G_0 = K_3 \cup K_3 \cup K_3,$$

where the union means that the three graphs are disjoint. Equivalently, graph theorists may write $G_0=3K_3$.

Each copy of $K_3$ contributes 3 vertices and 3 edges. Therefore,

$$|V(G_0)|=3\cdot 3=9, \qquad |E(G_0)|=3\cdot 3=9.$$

Every vertex is adjacent to the other two vertices in its own triangle, so every vertex has degree 2. No vertex in one component is adjacent to a vertex in another.

Common properties

Property	Value	Reason
Number of vertices	9	Three components with 3 vertices each
Number of edges	9	Three components with 3 edges each
Number of connected components	3	The triangles are disjoint
Degree of each vertex	2	Each vertex meets the other two in its triangle
Clique number	3	Each component is a 3-vertex clique
Chromatic number	3	Each triangle needs three different colors
Independence number	3	At most one vertex can be chosen from each triangle

The graph is not connected, because there is no path between vertices in different triangles. It is nevertheless planar: each triangle can be drawn separately without crossings.

Worked example: find the cycle rank

For a finite undirected graph, the cycle rank (also called the cyclomatic number) is

$$\mu = |E|-|V|+c,$$

where $c$ is the number of connected components. For this graph,

$$\mu=9-9+3=3.$$

This matches the picture: each triangle contains one independent cycle, and there are three separate triangles. The formula also works for disconnected graphs as long as you include the component count $c$.

Another way to describe the graph is with an adjacency matrix. If the vertices are ordered triangle by triangle, the matrix has three diagonal blocks, each equal to the adjacency matrix of $K_3$, and zeros everywhere else. Each block is

$$\begin{pmatrix} 0&1&1\\ 1&0&1\\ 1&1&0 \end{pmatrix}.$$

The zero entries between blocks record that different components have no edges between them.

Why the component structure helps

Many graph properties can be found by considering each triangle separately. For example, a proper coloring of one triangle needs three colors, and the same three colors can be reused in the other two components. Thus the chromatic number of the whole graph is 3, not 9.

An independent set contains no adjacent vertices. Within a triangle, it can include at most one vertex. Choosing one vertex from each component gives an independent set of size 3, so the independence number is 3. There are $3^3=27$ such maximum independent sets: three choices in each of the three components.

A spanning tree, by contrast, exists only for a connected graph. Since this graph has three components, it has no single spanning tree. It does have spanning forests that contain a tree in each component.

Common mistakes

• Calling the graph connected: Each triangle is connected on its own, but the union of three disjoint triangles has three components.
• Counting only the vertices: A triangle has 3 edges as well as 3 vertices, so the full graph has 9 of each.
• Assuming geometric shape matters: A triangle component is a 3-cycle in the graph-theoretic sense; its drawing need not have equal sides or even look like a triangle.
• Using a connected-graph formula without adjustment: For example, the cycle-rank formula must include $c=3$ here.

If a particular quantity is intended—such as the number of colorings, spanning forests, or automorphisms—the three-component description is enough to calculate it, but the requested quantity needs to be specified.