Related questions with answers

Three identical positive point charges are located at fixed points in space. Then charge $q_2$ is moved from its initial location to a final location as shown in the figure. Four different paths, marked (a) through (d), are shown. Path (a) follows the phortest line; path (b) takes $q_2$ around $q_3$ ; path (c) takes $q_2$ around $q_3$ and $q_1 ;$ path (d) takes $q_2$ out to infinity and then to the final location. Which path requires the least work? a) path (a) b) path (b) c) path (c) d) path (d) e) Thework is the same for all the paths.

You are playing air hockey with a friend. The puck is sitting at rest in his goal when he suddenly lifts his end of the table by 0.50 m. The puck slides down the tilted surface into your goal, 2.4 m away. Ignore friction. (a) How many seconds does it take the puck to reach your goal? (b) With what speed does the puck hit your goal?

(a) Identify the antagonist.

(b) What is the antagonist's goal?

Question

A closed path is a path from v to v. Show that a connected graph G is bipartite if and only if every closed path in G has even length.

Solution

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Step 1

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Given: A $\textbf{closed path}$ is a path from $v$ to $v$

A $\textbf{bipartite graph}$ is a simple graph whose vertices can be partitioned into two sets $V_1$ and $V_2$ such that there are no edges among the vertices of $V_1$ and no edges among the vertices of $V_2$ , while there can be edges between a vertex of $V_1$ and a vertex of $V_2$ .

To proof: A connected graph $G$ is bipartite if and only if every closed path in $G$ has even length.

$\textbf{PROOF}$

We will prove that "A connected graph $G$ is bipartite if and only if every closed path in $G$ has even length" by showing that every closed path in $G$ has even length when a connected graph $G$ is bipartite and showing that a connected graph $G$ is bipartite when every closed path in $G$ has even length.