Regular-graph case of the Nurdin–Baskoro–Salman–Gaos conjecture

Determine whether every d-regular graph G on n vertices satisfies the equality tvs(G) = (n+d)/(d+1), equivalently whether the counting lower bound for the total vertex irregularity strength of regular graphs is always attained.

Background

For a d-regular graph on n vertices, every vertex weight in a total k-labeling is the sum of d+1 labels, so distinct vertex weights yield the lower bound tvs(G) ≥ (n+d)/(d+1). The Nurdin–Baskoro–Salman–Gaos conjecture asserts that this bound is attained for every graph when specialized to regular graphs.

The paper proves the equality for all cubic and 4-regular graphs, and proves it for sufficiently large d-regular graphs for each fixed d≥2. The unrestricted regular-graph problem therefore remains unresolved, in particular for fixed degrees d≥5.