A POLYNOMIAL-TIME ALGORITHM FOR FINDING TOTAL COLORINGS OF PARTIAL k-TREES

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. Many combinatorial problems can be efficiently solved for partial k-trees, that is, graphs of treewidth bounded by a constant k. However, no polynomial-time algorithm has been known for the problem of finding a total coloring of a given partial k-tree with the minimum number of colors. This paper gives such a first polynomial-time algorithm.

Total Colorings Of Degenerate Graphs

“…Therefore by Lemma 2.1(c) we have χ t (G) ≤ ∆(G)+ k + 1< 5k + 4= O (1). A dynamic programming algorithm in [8] finds a total coloring of a partial k-tree G with χ t = χ t (G) colors in time O(nχ 2 4(k+1) t ). Since χ t (G) = O(1) and k = O(1), the algorithm takes time O(n) for this case.…”

“…However, no efficient algorithm has been known until now for the total coloring problem on partial k-trees. Although the total coloring problem can be solved in polynomial time for partial k-trees by a dynamic programming algorithm, the time complexity O(n 1+2 4(k+1) ) is very high [8].…”

Fixed-Parameter Tractability of Treewidth and Pathwidth

“…Such algorithms have been found for many combinatorial problems (see e.g., [8,9,33,94,138,144]), and also have been employed for problems from computational biology (see e.g., [100]), constraint satisfaction (see e.g., [40,47,78,94]), and probabilistic networks (see [99]). See e.g., also [3,2,6,12,32,39,38,42,50,70,79,80,82,85,88,106,105,108,145]. In other words: many graph problems become fixed-parameter tractable when parameterized by the treewidth of the input graph.…”