# A Linear Algorithm for Finding Total Colorings of Partial *k-Trees*

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## Abstract

A total coloring of a graph *G* is a coloring of all elements of *G*, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. The total coloring problem is to find a total coloring of a given graph with the minimum number of colors. Many combinatorial problems can be efficiently solved for partial *k-trees*, i.e., graphs with bounded tree-width. However, no efficient algorithm has been known for the total coloring problem on partial although a polynomial-time algorithm of very high order has been known. In this paper, we give a linear-time algorithm for the total coloring problem on partial *k*-trees with bounded *k*.

## Keywords

Linear Time Generalize Coloring Dynamic Programming Algorithm Linear Algorithm Cient Algorithm
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© Springer-Verlag Berlin Heidelberg 1999