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. 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 sufficiently large, say Δ ≥ 4s+3. Our proof yields an efficient 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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. Arnborg and J. Lagergren, Easy problems for tree-decomposable graphs, J. Algorithms, 12(2), pp. 308–340, 1991.
H. L. Bodlaender, Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees, J. Algorithms, 11(4), pp. 631–643, 1990.
O. V. Borodin, A. V. Kostochka and D. R. Woodall, List edge and list total colourings of multigraphs, J. Combinatorial Theory, Series B, 71, pp. 184–204, 1997.
R. B. Borie, R. G. Parker and C. A. Tovey, Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families, Algorithmica, 7, pp. 555–581, 1992.
R. Cole, K. Ost and S. Schirra, Edge-coloring bipartite multigraphs in O(E logD) time, Combinatorica, 21, pp. 5–12, 2001.
R. Diestel, Graph Theory, Springer, New York, 1997.
S. Isobe, X. Zhou and T. Nishizeki, A polynomial-time algorithm for finding total colorings of partial k-trees, Int. J. Found. Comput. Sci., 10(2), pp. 171–194, 1999.
T. R. Jensen and B. Toft, Graph Coloring Problems, John Wiley & Sons, New York, 1995.
D. Matula and L. Beck, Smallest-last ordering and clustering and graph coloring algorithms, J. Assoc. Comput. Mach., 30, pp. 417–427, 1983.
T. Nishizeki and N. Chiba, Planar Graphs: Theory and Algorithms, North-Holland, Amsterdam, 1988.
A. Sánchez-Arroyo. Determining the total colouring number is NP-hard, Discrete Math., 78, pp. 315–319, 1989.
G. Szekeres and H. Wilf, An inequality for the chromatic number of a graph, J. Combinatorial Theory, 4, pp. 1–3, 1968.
V. G. Vizing, Critical graphs with given chromatic class (in Russian), Metody Discret Analiz., 5, pp. 9–17, 1965.
H. P. Yap, Total Colourings of Graphs, Lect. Notes in Math., 1623, Springer, Berlin, 1996.
X. Zhou, S. Nakano and T. Nishizeki, Edge-coloring partial k-trees, J. Algorithms, 21, pp. 598–617, 1996.
X. Zhou and T. Nishizeki, Edge-coloring and f-coloring for various classes of graphs, J. Graph Algorithms and Applications, http://www.cs.brown.edu/publications/jgaa/, 3(1), pp. 1–18, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Isobe, S., Zhou, X., Nishizeki, T. (2001). Total Colorings of Degenerated Graphs. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds) Automata, Languages and Programming. ICALP 2001. Lecture Notes in Computer Science, vol 2076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48224-5_42
Download citation
DOI: https://doi.org/10.1007/3-540-48224-5_42
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42287-7
Online ISBN: 978-3-540-48224-6
eBook Packages: Springer Book Archive