An eccentric coloring of trees

Eccentric coloring is a new variation of coloring, where higher numbered colors can not be used as freely as lower numbered colors. In addition there is a correspondence between the eccentricity (max distance) of a vertex and the highest legal color for that vertex. In this note we investigate eccentric coloring of trees. We give the eccentric chromatic number or a bound on the eccentric chromatic number for several simple classes of trees. In particular we show the eccentric chromatic number for paths (χe = 3), spiders (χe = 3) and caterpillars (χe ≤ 7). Further, we discuss the eccentric chromatic number of complete k-ary trees and show that the complete binary trees have eccentric chromatic number χe ≤ 7. We also show that large binary trees are eccentrically colorable and have χe ≤ 7. We then conclude by showing that no complete k-ary tree, k ≥ 3, is eccentrically colorable.