A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is sdegenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤ s. We prove that an s-degenerate graph G has a total coloring with ∆+1 colors if the maximum degree ∆ of G is sufficiently large, say ∆ ≥ 4s+3. Our proof yields an efficient algorithm to find such a total coloring. We also give a lineartime algorithm to find a total coloring of a graph G with the minimum number of colors if G is a partial k-tree, that is, the tree-width of G is bounded by a fixed integer k.
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.
SUMMARYIn this paper, we investigate a relationship between the length-decreasing self-reducibility and the many-one-like reducibilities for partial multivalued functions. We show that if any parsimonious (manyone or metric many-one) complete function for NPMV (or NPMV g ) is length-decreasing self-reducible, then any function in NPMV (or NPMV g ) has a polynomial-time computable refinement. This result implies that there exists an NPMV (or NPMV g )-complete function which is not lengthdecreasing self-reducible unless P = NP.
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