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Chapter 11: Problem 4

Prove that any subgraph of a bipartite graph is bipartite.

Short Answer

Expert verified
The proof rests on the definition of a subgraph and a bipartite graph. Since a bipartite graph has its vertices divided into two disjoint sets and any subgraph will contain vertices and edges from these pre-established sets, any subgraph will also be bipartite.

Step by step solution

01

Definition of Bipartite Graph and Subgraph

Firstly, recall the definition of a bipartite graph and define what a subgraph is. A graph is said to be bipartite if it's vertices can be split into two disjoint sets \(U\) and \(V\) such that every edge connects a vertex in \(U\) to one in \(V\). A subgraph \(S\) of a graph \(G\) is a graph whose vertices and edges are all in \(G\).
02

Extracting a Subgraph

Suppose we have a bipartite graph \(G\) with the disjoint vertex sets \(U\) and \(V\), and one extracts a subgraph \(S\) from this graph. Now, this subgraph \(S\) will have some, possibly all, vertices of \(G\), and some, possibly all, edges of \(G\). The vertices can still be labeled as belonging to either \(U\) or \(V\).
03

Analysis of Edges in Subgraph

Since only edges already existing in the bipartite graph \(G\) have been selected to form the subgraph \(S\), it can be stated that all selected edges for \(S\) still connect vertices from the disjoint sets \(U\) and \(V\). No additional edges have been created in the subgraph that connect vertices from the same set.
04

Conclusion

Hence, due to the fact that the edges still only connect vertices across \(U\) and \(V\), it can be concluded that the subgraph \(S\) of the bipartite graph \(G\) is also bipartite.

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Most popular questions from this chapter

Prove that in any directed graph or multigraph \(G=(V, E), \sum_{\text {uev }} o d(v)=\sum_{\text {revid }}(v)\).

a) Let \(G=(V, E)\) be a connected bipartite undirected graph with \(V\) partitioned as \(V_{1} \cup V_{2}\). Prove that if \(\left|V_{1}\right| \neq\left|V_{2}\right|\), then \(G\) cannot have a Hamilton cycle. b) Prove that if the graph \(G\) in part (a) has a Hamilton path, then \(\left|V_{1}\right|-\left|V_{2}\right|=\pm 1\). c) Give an example of a connected bipartite undirected graph \(G=(V, E)\), where \(V\) is partitioned as \(V_{1} \cup V_{2}\) and \(\left|V_{1}\right|=\left|V_{2}\right|-1\), but \(G\) has no Hamilton path.

If \(n \geq 3\), how many different Hamilton cycles are there in the wheel graph \(W_{n} ?\)

a) How many vertices and how many edges are there in the complete bipartite graphs \(K_{4,7}, K_{7,11}\), and \(K_{m, n}\) where \(m, n \in \mathbf{Z}^{+}\)? b) If the graph \(K_{m, 12}\) has 72 edges, what is \(m\) ?

Let \(G=(V, E)\) be a connected undirected graph. a) What is the largest possible value for \(|V|\) if \(|E|=19\) and \(\operatorname{deg}(v) \geq 4\) for all \(v \in V\) ? b) Draw a graph to demonstrate each possible case in part (a).
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