Improved upper bounds for λ-backbone colorings along matchings and stars

Haitze J. Broersma, L. Marchal, Daniël Paulusma, M. Salman

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

5 Citations (Scopus)

Abstract

We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph $G = (V,E)$ and a spanning subgraph $H$ of $G$ (the backbone of $G$), a $\lambda$-backbone coloring for $G$ and $H$ is a proper vertex coloring $V\to\{1,2,\ldots\}$ of $G$ in which the colors assigned to adjacent vertices in $H$ differ by at least $\lambda$. The main outcome of earlier studies is that the minimum number $\ell$ of colors for which such colorings $V\to\{1,2,\ldots, \ell\}$ exist in the worst case is a factor times the chromatic number (for all studied types of backbones). We show here that for split graphs and matching or star backbones, $\ell$ is at most a small additive constant (depending on $\lambda$) higher than the chromatic number. Despite the fact that split graphs have a nice structure, these results are difficult to prove. Our proofs combine algorithmic and combinatorial arguments. We also indicate other graph classes for which our results imply better upper bounds on $\ell$ than the previously known bounds.
Original languageUndefined
Title of host publicationProceedings of SOFSEM 2007: Theory and Practice of Computer Science
EditorsJ. van Leeuwen, G.F. Italiano, W. van der Hoek, C. Meinel, H. Sack, F. Plásil
Place of PublicationBerlin
PublisherSpringer
Pages188-199
Number of pages12
ISBN (Print)978-3-540-69506-6
DOIs
Publication statusPublished - 13 Jul 2007

Publication series

NameLecture Notes in Computer Science
PublisherSpringer Verlag
NumberLNCS4549
Volume4362
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Keywords

  • EWI-11097
  • IR-61929
  • METIS-241921

Cite this

Broersma, H. J., Marchal, L., Paulusma, D., & Salman, M. (2007). Improved upper bounds for λ-backbone colorings along matchings and stars. In J. van Leeuwen, G. F. Italiano, W. van der Hoek, C. Meinel, H. Sack, & F. Plásil (Eds.), Proceedings of SOFSEM 2007: Theory and Practice of Computer Science (pp. 188-199). [10.1007/978-3-540-69507-3] (Lecture Notes in Computer Science; Vol. 4362, No. LNCS4549). Berlin: Springer. https://doi.org/10.1007/978-3-540-69507-3_15, https://doi.org/10.1007/978-3-540-69507-3