Because g is odd, the labeling is also odd: . Hence, by Tucker's lemma, there are two adjacent vertices with opposite labels. Assume w.l.o.g. that the labels are . By the definition of l, this means that in both and , coordinate #1 is the largest coordinate: in this coordinate is positive while in it is negative. By the construction of the triangulation, the distance between and is at most , so in particular (since and have opposite signs) and so . But since the largest coordinate of is coordinate #1, this means that for each . So , where is some constant depending on and the norm which you have chosen.
Because g is odd, the labeling is also odd: . Hence, by Tucker's lemma, there are two adjacent vertices with opposite labels. Assume w.l.o.g. that the labels are . By the definition of l, this means that in both and , coordinate #1 is the largest coordinate: in this coordinate is positive while in it is negative. By the construction of the triangulation, the distance between and is at most , so in particular (since and have opposite signs) and so . But since the largest coordinate of is coordinate #1, this means that for each . So , where is some constant depending on and the norm which you have chosen.