Total Coloring of Block-Cactus Graphs

In this paper, we present new results about the coloring of graphs. We generalize the notion of proper vertex-coloring, introducing the concept of range-coloring of order \( k \). The relation between range-coloring of order \( k \) and total coloring is presented: we show that for any graph \( G \) that has a range-coloring of order \( \Delta(G) \) with \( t \) colors, there is a total coloring of \( G \) that uses \( (t+1) \) colors. This result provides a framework to prove that some families of graphs satisfy the total coloring conjecture. We exemplify with the family of block-cactus graphs.