IWDC '02 Proceedings of the 4th International Workshop on Distributed Computing, Mobile and Wireless Computing
Discrete Applied Mathematics
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The span $\la (G)$ 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 $\{0,1, \ldots, k \}$ 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 $\{0,1, \ldots, \la (G) \}$ and no others. We prove that all trees except stars are full-colorable. The connected graph $G$ with the smallest number of vertices exceeding $\la (G)$ that is not full-colorable is $C_6$. 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.