Informally ...

Given a network, or graph, an Euler Cycle is a tour from vertex to vertex, finishing where it started, and traversing every edge exactly once.

In contrast:

• An Euler Path traverses every edge but need not finish where it started;
• A Hamilton Cycle visits every vertex exactly once and finishes where it started
  • ... so the start/end vertex is, in some sense, visited twice
    • ... except that starting at a vertex is not regarded as "a visit"; and
• A Hamilton Path includes every vertex exactly once,
  • ... and need not finish where it started.

More formally ...

Given a graph $G=(V,E)$ where $|E|=m,$ an Euler Cycle is a list $L=[v_0,v_1,...,v_m]$ of vertices such that:

• $v_0=v_m$
  • That is, the cycle ends where it started

• For every $i\in{1,2,...,m},$ $(v_{i-1},v_i)\in{E}$
  • That is, going from vertex to vertex is along an edge

• For all $e=(u,v)\in{E},$ there exists exactly one $i$ such that either $u=v_i$ and $v=v_{i+1}$ or vice versa.
  • That is, every edge is traversed exactly once.

The "degree" of a vertex is the number of edges incident to it, and it is a well-known theorem that a connected graph is Eulerian if and only if every vertex has even degree. A graph has an Eulerian Path that is not a cycle if and only if every vertex has even degree except for two.

• https://en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg