Sum coloring of distance and circulant graphs

A distance graph (resp. circulant graph) is a graph obtained from the path (resp. the cycle) of order n by adding chordal edges between any pair of vertices at distance d∈D, with D⊂Zn. The chromatic sum of a graph is the smallest sum of colors among all proper colorings of this graph. This paper studies the chromatic sum for some classes of distance graphs and circulant graphs depending on the value of D.