Question
Suppose a graph G has at least one edge. Prove that the chromatic number of G is 2 if and only if G is bipartite.
168
likes840 views
Answer to a math question Suppose a graph G has at least one edge. Prove that the chromatic number of G is 2 if and only if G is bipartite.
Gerhard
4.592 Answers
To prove that the chromatic number of G is 2 if and only if G is bipartite, we will show both implications:
1. If the chromatic number of G is 2, then G is bipartite:
If the chromatic number of G is 2, then we can color the vertices of G with only 2 colors. This implies that the graph can be partitioned into 2 disjoint sets such that each edge of the graph connects vertices from different sets. Therefore, G is bipartite.
2. If G is bipartite, then the chromatic number of G is 2:
If G is bipartite, we can partition the vertices of G into 2 disjoint sets such that each edge of the graph connects vertices from different sets. Since no two vertices within the same set are adjacent, we can color the vertices in one set with one color and the vertices in the other set with a different color. This demonstrates that the chromatic number of G is 2.
Therefore, we have shown both implications:
"If the chromatic number of G is 2, then G is bipartite" and "If G is bipartite, then the chromatic number of G is 2", which completes the proof.
\textbf{Answer:} The chromatic number of G is 2 if and only if G is bipartite.
1. If the chromatic number of G is 2, then G is bipartite:
If the chromatic number of G is 2, then we can color the vertices of G with only 2 colors. This implies that the graph can be partitioned into 2 disjoint sets such that each edge of the graph connects vertices from different sets. Therefore, G is bipartite.
2. If G is bipartite, then the chromatic number of G is 2:
If G is bipartite, we can partition the vertices of G into 2 disjoint sets such that each edge of the graph connects vertices from different sets. Since no two vertices within the same set are adjacent, we can color the vertices in one set with one color and the vertices in the other set with a different color. This demonstrates that the chromatic number of G is 2.
Therefore, we have shown both implications:
"If the chromatic number of G is 2, then G is bipartite" and "If G is bipartite, then the chromatic number of G is 2", which completes the proof.
\textbf{Answer:} The chromatic number of G is 2 if and only if G is bipartite.
Frequently asked questions (FAQs)
What is the number of ways to arrange 4 books on a shelf, considering the order matters?
+
What is the missing side length in a right triangle if one leg measures 5 units and the hypotenuse measures 13 units?
+
Question: What is the formula to find the surface area of a cube, given that the length of each side is represented by 's'?
+
New questions in Mathematics