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Chapter 11: Problem 10

Can a bipartite graph contain a cycle of odd length? Explain.

Short Answer

Expert verified
No, a bipartite graph cannot contain a cycle of odd length.

Step by step solution

01

Understanding Bipartite Graphs

A graph \(G\) is said to be bipartite if its vertex set \(V\) can be partitioned into two non-empty sets \(V1\) and \(V2\) such that every edge in the graph connects a vertex in \(V1\) to a vertex in \(V2\). That is, there are no edges that connect vertices within the same set.
02

Understanding Cycles

In graph theory, a cycle is a path of edges and vertices wherein a vertex is reachable from itself. The length of the cycle is considered as the number of edges in the cycle.
03

Bipartite Graph and Cycles

Now considering a cycle of odd length in a bipartite graph and let's assume that one such cycle exists. If you start at one vertex, each move along an edge in the cycle changes the partition that you're in. Therefore, to end up in the same partition you started in (i.e., at the same vertex), you must make an even number of moves. Therefore, any cycle in a bipartite graph must consist of an even number of edges 鈥� i.e., the cycle must be of even length.
04

Conclusion

Thus, a bipartite graph cannot contain a cycle of odd length.

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Most popular questions from this chapter

Let \(m, n \in \mathbf{Z}^{+}\)with \(m \leq n\). Under what condition(s) on \(m, n\) will every edge in \(K_{m, n}\) be in exactly one of two isomorphic subgraphs of \(K_{m, n} ?\)

Let \(G=(V, E)\) be a loop-free connected undirected graph with \(|V| \geq 2\). Prove that \(G\) contains two vertices \(v, w\), where \(\operatorname{deg}(v)=\operatorname{deg}(w)\)

a) For \(n \geq 3\), how many different Hamilton cycles are there in the complete graph \(K_{n}\) ? b) How many edge-disjoint Hamilton cycles are there in \(K_{21} ?\) c) Nineteen students in a nursery school play a game each day where they hold hands to form a circle. For how many days can they do this with no student holding hands with the same playmate twice?

Determine whether or not the loop-free undirected graphs with the following incidence matrices are isomorphic. a) \(\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0\end{array}\right],\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right]\) b) \(\left[\begin{array}{llll}1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\\ 0 & 0 & 0 & 1\end{array}\right],\left[\begin{array}{llll}1 & 0 & 0 & 1 \\\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1\end{array}\right]\) c) \(\left[\begin{array}{lllll}1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0\end{array}\right],\left[\begin{array}{lllll}1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 & 0\end{array}\right]\)

Let \(G=(V, E)\) be a loop-free undirected graph. Prove that if \(G\) contains no cycle of odd length, then \(G\) is bipartite.
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