Mathematics — Graph Theory · alternating-eulerian-circuits
Problem
Let G be a connected multigraph. Assume that each edge of G has been colored either red or blue. For any vertex v of G, we let degrv denote the number of red edges containing v (where red loops are counted twice), and we let degbv denote the number of blue edges containing v (where blue loops are counted twice). A walk w of G is said to be alternating if no two consecutive edges of w have the same color (i.e., if its edges alternate between red and blue, starting with either red or blue). An alternating walk w of G is said to be fully alternating if the first and the last edges of w have different colors (or else if w has no edges).
Prove that the multigraph G has a fully alternating Eulerian circuit if and only if each vertex v of G satisfies degrv=degbv.