Maximising 1’s through proper labellings

Abstract

We investigate graph proper labellings, i.e., assignments of labels to the edges so that no two adjacent vertices are incident to the same sum of labels, with the additional requirement that label 1 must be assigned to as many edges as possible. The study of such objects is motivated by practical concerns, and by connections with other types of proper labellings in which other additional properties (such as minimising the sum of assigned labels, or minimising the use of label 3) must be met. We prove that maximising 1's is a problem on its own, in that it is not equivalent to any of these other labelling problems with optimisation concerns. We then provide labelling tools and techniques for designing proper labellings with many 1's. As a result, we prove that, for several graph classes, it is always possible to design proper labellings where label 1 is assigned to about half the edges.