How can I tell if this graph is 4-Edge-Connected? Is this graph 4-Edge Connected? What is the way to determine?

 A: A graph is $4$-edge-connected if deleting fewer than four edges cannot disconnect it.  For your graph:


*

*Deleting one edge cannot disconnect it because this graph has a Hamiltonian cycle, $(1,2,5,7,8,10,9,6,3,4)$.  Deleting an edge not on the cycle does not disconnect the graph.  Deleting an edge on the cycle does not disconnect the graph (go the other way around the cycle).

*There are a pair of edges whose deletion disconnects the graph: $\{3,6\}$ and $\{5,7\}$.


Notice that the subgraph on vertices $\{1,2,3,4\}$ and also on the vertices $\{7,8,9,10\}$ are $K_4$s, so are $4$-edge-connected.  Temporarily mentally collpase them to points.  The resulting four vertex multigraph is a cycle with edge multiplicities (in order going around the cycle) $1$, $2$, $1$, and $2$.  From this we can see that deleting the two multiplicty-$1$ edges will disconnect the reduced graph, hence disconnect the original graph.
A general method is, for each pair of vertices $a,b$, find at least $k-1$ edge-disjoint paths from $a$ to $b$.  For instance, in your graph, $(1,2)$, $(1,3,2)$, and $(1,4,2)$ are three paths from vertex $1$ to vertex $2$ and no edge appears in two (or more) of these paths.  So to separate vertex $1$ from vertex $2$ requires deleting at least three edges.  Continue for every other pair of vertices.
There exist algorithms that compute variants of maxflow to find the largest $k$ for which a given graph is $k$-edge-connected.
