Let P be a collection of nontrivial simple paths in a tree T. The edge intersection graph of P, denoted by EPT(P), has vertex set that corresponds to the members of P, and two vertices are joined by an edge if the corresponding members of P share a common edge in T. An undirected graph G is called an edge intersection graph of paths in a tree, if G = EPT(P) for some P and T. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs.