Balanced valuations and flows in multigraphs

Abstract

A balanced valuation of a multigraph H H is a mapping w w of its vertex-set V ( H ) V(H) into R R such that ∀ S ⊆ V ( H ) \forall S \subseteq V(H) the number of edges of H H with exactly one vertex in S S is greater than or equal to | ∑ v ∈ S w ( v ) | |{\sum _{v \in S}}w(v)| ; 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 3 if and only if it has a balanced valuation with values in { − 2 , + 2 } \{ - 2, + 2\} (Bondy [5]). Every planar cubic 2 2 -edge connected multigraph has a balanced valuation with values in { − 5 / 3 , + 5 / 3 } \{ - 5/3, + 5/3\} . Every planar 5 5 -regular 4 4 -edge connected multigraph has a balanced valuation with values in { − 3 , + 3 } \{ - 3, + 3\} .