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Eulerian paths on graphs with one odd vertex must end and begin at that vertex.
Eulerian paths on graphs with two odd vertices must end and begin and one odd vertex and end at the other.
Eulerian paths don't exist on graphs with more than two odd vertices.
Graphs with an odd number of odd vertices do not exist.

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Circuit is a path that goes starts and ends at a same vertex or point.
This is kinda like an electrical circuit

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What did Euler do to find the groundwork of Graph Theory?

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What did Euler do to find the groundwork of Graph Theory?

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Euler had been visiting a town named Königsberg in Prussia, when he was confronted with the problem that is now known as  the Königsberg Bridge Problem. This is what got him thinking about what is now called graph theory.

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If an edge connects two vertices than that edge is an incident to those two vertices. The degree of a vertex is the number of edges that are incident to that vertex. So, if  edges are incident to a vertex, then that vertex is an even vertex. If three edges are incident to a vertex, then that vertex is an odd vertex.

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A bipartite graph is a graph whose the vertices can be separated into two sets so that each line/edge on the graph can connect a vertex in the first set to a vertex in the second.

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Eulerian Path is a path that cover all of the edges or paths only once.
If you look at the graph besides this and see the numbers as steps, you can see that the path that it makes a path without going over the same path once

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Graphs are complete when each pair of vertices is connected by exactly one edge.

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if all of the vertices on a graph are even, this graph is an Eulerian circuit.

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Paths travel along edges and vertices to connect a pair of vertices together.

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Eulerian paths and Circuits

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Many graphs can be drawn on a piece of paper so that the edges intersect only at the vertices. This kind of graph is called a planar graph. Non-planar graphs have edges that intersect basically anywhere on the graphs, not just on the vertices.

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Unlike Eulerian Circuits, Hamiltonian Circuits focus on trying to visit each vertex exactly once, and can have edges traveled multiple times. While Eulerain Circuits and Paths have characteristics that can help one solve a Eulerian Problem, Hamiltonian paths do not. This makes them more difficult to find as one usually must find them through luck, or trail and error.

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A graph is made of vertices and the edges that connect vertices to each other.

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When was Graph Theory created (or known to officially exist)?

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When was Graph Theory created (or known to officially exist)?

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Graph Theory was first known to "officially exist" in the mid-1800's

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Leonard Euler is the father of graph theory