Full Color Theorems for L(2, 1)-Colorings

The span λ(G) of a graph G is the smallest k for which G’s vertices can be L(2, 1)-colored, i.e., colored with integers in {0, 1, . . . , k} so that adjacent vertices’ colors differ by at least two, and colors of vertices at distance two differ. G is full-colorable if some such coloring uses all colors in {0, 1, . . . , λ(G)} and no others. We prove that all trees except stars are full-colorable. The connected graph G with the smallest number of vertices exceeding λ(G) that is not full-colorable is C6. We describe an array of other connected graphs that are not full-colorable and go into detail on full-colorability of graphs of maximum degree four or less.