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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

a) Let \(k \in \mathbf{Z}^{+}, k \geq 3\). If \(G=(V, E)\) is a connected planar graph with \(|V|=v,|E|=e\), and each cycle of length at least \(k\), prove that \(e \leq\left(\frac{k}{k-2}\right)(v-2)\). b) What is the minimal cycle length in \(K_{3,3}\) ? c) Use parts (a) and (b) to conclude that \(K_{3,3}\) is nonplanar. d) Use part (a) to prove that the Petersen graph is nonplanar.

Let \(G=(V, E)\) be a loop-free undirected \(n\)-regular graph with \(|V| \geq 2 n+2\). Prove that \(\bar{G}\) (the complement of \(G\) ) has a Hamilton cycle.

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?

Prove that every loop-free connected planar graph has a vertex \(v\) with \(\operatorname{deg}(v)<6\).

Let \(G\) be a loop-free undirected graph where \(\Delta=\max _{v e v}\\{\) deg \((v)\\}\). a) Prove that \(\chi(G) \leq \Delta+1\). b) Find two types of graphs \(G\) where \(\chi(G)=\Delta+1\).
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