Question

Prove that a tree is a bipartite graph.

Short answer

A tree is bipartite because it can be colored with two colors without adjacent vertices sharing the same color.

Step-by-step solution

Understand the Definition of a Tree and Bipartite Graph

A tree is a connected graph with no cycles, while a bipartite graph is a graph whose vertices can be divided into two disjoint sets such that no two graph vertices within the same set are adjacent. We will use these definitions to prove that a tree is bipartite.

Choose a Starting Point

Select an arbitrary vertex in the tree as the starting point. This will act as the root of the tree for our purpose of proving it can be colored bipartitely.

Assign Colors to Vertices

Begin by coloring the starting vertex one color, say color A. Then color all adjacent vertices with the second color, color B. Continue this pattern: if a vertex is colored with color A, color all its adjacent vertices with color B, and vice versa.

Show No Adjacent Vertices Have the Same Color

Since a tree has no cycles, there is no possibility of two vertices that are on the same layer having edges connecting them. Thus, no two adjacent vertices painted with the same color can occur, respecting the rule of bipartition.

Conclude Based on Coloring Validity

You've shown that by alternating colors at each level of vertices from the starting point, the graph remains free of same-color adjacency. Therefore, the tree graph is bipartite because it satisfies the property of bipartition with disjoint vertex sets.

Key concepts

Graph Theory

Graph theory is a branch of mathematics that studies networks of interconnected points called vertices and lines connecting them called edges. It's an essential field for understanding various structures featuring pairwise connections. A graph can model anything from social networks to transportation systems.

In graph theory:

• A **vertex** is a node or a point
• An **edge** is a connection between two vertices
• Graphs can be directed or undirected, depending on whether the edges have a direction
• Graphs can have cycles (a closed loop) or be acyclic (like trees, which don't loop back)

Trees and bipartite graphs are specific types of graphs within this domain. Trees are connected graphs without cycles, and recognizing this is crucial in proving that a tree is a bipartite graph. Since trees are acyclic, they have properties that align well with bipartite conditions, thus making them an interesting subject in graph theory studies.

Vertex Coloring

Vertex coloring is a method of assigning labels or "colors" to the vertices of a graph such that no two adjacent vertices share the same color. This concept is central when discussing bipartite graphs, especially in proving how trees are inherently bipartite.

In practice, this involves:

• Choosing a starting vertex and assigning it the first color
• Coloring all directly connected vertices with a different color
• Alternating colors as you move through the graph levels

In a tree, which has no cycles, vertex coloring inherently respects the condition that adjacent vertices receive different colors. This systematic coloring ensures that the tree divides into a bipartite graph, confirming its structural properties.

Disjoint Sets

Disjoint sets are groups with no common elements between them. In the context of a bipartite graph, the graph's vertices can be split into two disjoint sets where each edge only connects a vertex from one set to a vertex in the other set. This separation ensures that no two vertices within the same set are connected directly.

When considering trees:

• Start with any vertex and consider it in the first set
• Place all adjacent vertices into the second set
• Continue this pattern, placing newly reachable vertices into the opposite set of their adjacent vertex

With no cycles present in a tree, this approach of creating disjoint sets will never fail. A tree's structural simplicity makes it naturally adaptable to forming these bipartite divisions.

Connected Graph

A connected graph is one where there's a path between every pair of vertices. In essence, you can traverse the entire graph without lifting your pen, starting anywhere and reaching any point. A tree is a perfect example of a connected, simple yet acyclic graph, ensuring connectivity without any loops.

Key properties include:

• Being a single connected component, meaning there's just one "piece" or unified whole
• No isolated vertices; everyone is part of the main structure
• An additional edge would introduce a cycle and disrupt the tree's acyclic nature

Since trees are connected graphs without cycles, they interestingly lie perfectly within the set of bipartite graphs, offering clear paths yet maintaining distinctive separability at each level of nodes.