RU-ALLRUSSIAN 1966 Problem 8

A graph has at least three vertices. Given any three vertices A, B, C of the graph we can find a path from A to B which does not go through C. Prove that we can find two disjoint paths from A to B.

[A graph is a finite set of vertices such that each pair of distinct vertices has either zero or one edges joining the vertices. A path from A to B is a sequence of vertices $\mathrm{A_1}$ , $\mathrm{A_2}$ , ..., $\mathrm{A_n}$ such that $\mathrm{A = A_1}$ , $\mathrm{B = A_n}$ and there is an edge between $\mathrm{A_i}$ and $\mathrm{A_{i + 1}}$ for $\mathrm{i} = 1$ , 2, ..., n- 1. Two paths from A to B are disjoint if the only vertices they have in common are A and B.]