Abstract
Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology, and the framework through which it was studied, we introduce the colinear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be colinearly colored in polynomial time by proposing a simple algorithm. The colinear coloring of a graph G is a vertex coloring such that two vertices can be assigned the same color, if their corresponding clique sets are associated by the set inclusion relation (a clique set of a vertex u is the set of all maximal cliques containing u); the colinear chromatic number λ(G) of G is the least integer k for which G admits a colinear coloring with k colors. Based on the colinear coloring, we define the χ-colinear and α-colinear properties and characterize known graph classes in terms of these properties.
This research is co-financed by E.U.-European Social Fund (75%) and the Greek Ministry of Development-GSRT (25%).
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Boesch, F.T., Gimpel, J.F.: Covering the points of a digraph with point-disjoint paths and its application to code optimization. J. of the ACM 24, 192–198 (1977)
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)
Brooks, R.L.: On colouring the nodes of a network. Proc. Cambridge Phil. Soc. 37, 194–197 (1941)
Chvátal, V., Hammer, P.L.: Aggregation of inequalities for integer programming. Ann. Discrete Math. I, 145–162 (1977)
Civan, Y., Yalçin, E.: Linear colorings of simplicial complexes and collapsing. J. Comb. Theory A 114, 1315–1331 (2007)
Csorba, P., Lange, C., Schurr, I., Wassmer, A.: Box complexes, neighborhood complexes, and the chromatic number. J. Comb. Theory A 108, 159–168 (2004)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980); annals of Discrete Mathematics, 2nd edn., vol. 57. Elsevier (2004)
Hopcroft, J., Karp, R.M.: A n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Computing 2, 225–231 (1973)
Kneser, M.: Aufgabe 300. Jahresbericht der Deutschen Mathematiker-Vereinigung 58, 2 (1955)
Lovász, L.: Kneser’s conjecture, chromatic numbers and homotopy. J. Comb. Theory A 25, 319–324 (1978)
Matoušek, J., Ziegler, G.M.: Topological lower bounds for the chromatic number: a hierarchy. Jahresbericht der Deutschen Mathematiker-Vereinigung 106, 71–90 (2004)
Nikolopoulos, S.D.: Recognizing cographs and threshold graphs through a classification of their edges. Inform. Proc. Lett. 74, 129–139 (2000)
Ziegler, G.M.: Generalised Kneser coloring theorems with combinatorial proofs. Inventiones mathematicae 147, 671–691 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ioannidou, K., Nikolopoulos, S.D. (2009). Colinear Coloring on Graphs. In: Das, S., Uehara, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2009. Lecture Notes in Computer Science, vol 5431. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00202-1_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-00202-1_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00201-4
Online ISBN: 978-3-642-00202-1
eBook Packages: Computer ScienceComputer Science (R0)