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An Euler circuit is one of the fundamental concepts in graph theory. It is essentially a path that traverses each edge of a graph exactly once and returns to the starting point. For a graph to possess an Euler circuit, every vertex must have an even degree. This means that each vertex should be connected to an even number of edges.

This necessity arises because if a vertex is part of an Euler circuit, every entrance to the vertex must be paired with an exit. A simple example is a triangular graph where each vertex connects to two others via three edges. Traversing the triangle, you start at a vertex, visit each edge exactly once, and return to your starting point, forming an Euler circuit.

Overall, understanding Euler circuits aids in solving problems related to efficiently covering all connections in practical applications, like routing problems.

The idea of connectivity is crucial in understanding graph structures. A connected graph is defined as a graph in which there is a path between every pair of vertices. In essence, no vertex stands isolated from the rest of the graph.

Imagine a network of friendships: if every friend is somehow connected to every other friend directly or indirectly, you have a connected graph. If even one friend is not connected, the graph becomes disconnected.

In context of Euler circuits, a connected graph ensures that the circuit can traverse through every edge without needing to "jump" to a separate, disconnected part of the graph. Thus, connectivity is key to ensuring an Euler circuit can exist.

Proof by contradiction is a classic approach in mathematics used to establish a statement's truth. The idea is straightforward: assume the opposite of what you want to prove, then show that this assumption leads to a contradiction, something logically impossible.

To understand how this applies to graphs with Euler circuits, imagine assuming that a graph with an Euler circuit is not connected (the opposite of what we want to prove). If this assumption leads to an impossibility—that the circuit cannot be completed without joining all parts of the graph—then the assumption must be false. Hence, the statement that a graph with an Euler circuit must be connected is true.

This method is not only powerful in graph theory but across various fields of mathematics, providing a strong tool for proofs.

Graph connectivity isn't just about ensuring a path between every vertex; it's about understanding how the graph's structure allows or prevents such paths.

In graph theory, ensuring connectivity implies studying the ways vertices and edges are juxtaposed and planning strategies for maintaining these connections. Key questions include: Where should new edges be placed to improve connectivity? What happens to connectivity if an edge or vertex is removed?

Particularly in relation to Euler circuits, connectivity ensures that there is no division in the graph that leaves any edge untouched or unreachable. When a graph is fully connected, you can freely traverse it using different paths, vital for configurations requiring maximum coverage, like network designs and route planning.