MathDB

We are given a finite, simple, non-directed graph. Ann writes positive real numbers on each edge of the graph such that for all vertices the following is true: the sum of the numbers written on the edges incident to a given vertex is less than one. Bob wants to write non-negative real numbers on the vertices in the following way: if the number written at vertex

v

v_0

, and Ann's numbers on the edges incident to

v

are

e_1,e_2,\ldots,e_k

, and the numbers on the other endpoints of these edges are

v_1,v_2,\ldots,v_k

, then

v_0=\sum_{i=1}^k e_iv_i+2022

. Prove that Bob can always number the vertices in this way regardless of the graph and the numbers chosen by Ann.
Proposed by Boldizsár Varga, Verőce

Problem Statement