r-Dynamic Chromatic Number of Some Line Graphs

An r -dynamic coloring of a graph G is a proper coloring c of the vertices such that | c ( N ( v ))| ≥ min { r , deg( v )}, for each v ∈ V ( G ). The r -dynamic chromatic number of a graph G is the smallest k such that G admits an r -dynamic coloring with k colors. In this paper, we obtain the r -dynamic chromatic number of the line graph of helm graphs H n for all r between minimum and maximum degree of H n . Moreover, our proofs are constructive, what means that we give also polynomial time algorithms for the appropriate coloring. Finally, as the first, we define an equivalent model for edge coloring.