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A balanced valuation of a multigraph H is a mapping w of its vertex-set V(H) into R such that VS C V(H) the number of edges of H with exactly one vertex in S is greater than or equal to IXvEsw(v)1; we apply the theory of flows in networks to obtain known and new results on balanced valuations such as: A cubic multigraph has chromatic index 3 if and only if it has a balanced valuation with values in { -2,+2) (Bondy [5]). Every planar cubic 2-edge connected multigraph has a balanced valuation with values in (-5/3,+ 5/3). Every planar 5-regular 4-edge connected multigraph has a balanced valuation with values in {-3, + 3).

Balanced valuations and flows in multigraphs