Determining the total colouring number is NP-hard
Discrete Mathematics
Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees
Journal of Algorithms
Easy problems for tree-decomposable graphs
Journal of Algorithms
Journal of Algorithms
List edge and list total colourings of multigraphs
Journal of Combinatorial Theory Series B
Smallest-last ordering and clustering and graph coloring algorithms
Journal of the ACM (JACM)
L(2,1)-labelings of the edge-path-replacement of a graph
Journal of Combinatorial Optimization
L(d,1)-labelings of the edge-path-replacement of a graph
Journal of Combinatorial Optimization
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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. A graph G is s-degenerated 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-degenerated graph G has a total coloring with Δ + 1 colors if the maximum degree Δ of G is suffciently large, say Δ ≥ 4s+3. Our proof yields an effcient algorithm to find such a total coloring. We also give a linear-time algorithm to find a total coloring of a graph G with the minimum number of colors if G is a partial k-tree, i.e. the tree-width of G is bounded by a fixed integer k.