# 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 language Undefined Proceedings of SOFSEM 2007: Theory and Practice of Computer Science J. van Leeuwen, G.F. Italiano, W. van der Hoek, C. Meinel, H. Sack, F. Plásil Berlin Springer 188-199 12 978-3-540-69506-6 Published - 13 Jul 2007

### Publication series

Name Lecture Notes in Computer Science Springer Verlag LNCS4549 4362 0302-9743 1611-3349

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

Improved upper bounds for λ-backbone colorings along matchings and stars. / Broersma, Haitze J.; Marchal, L.; Paulusma, Daniël; Salman, M.

Proceedings of SOFSEM 2007: Theory and Practice of Computer Science. ed. / J. van Leeuwen; G.F. Italiano; W. van der Hoek; C. Meinel; H. Sack; F. Plásil. Berlin : Springer, 2007. p. 188-199 10.1007/978-3-540-69507-3 (Lecture Notes in Computer Science; Vol. 4362, No. LNCS4549).

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

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T1 - Improved upper bounds for λ-backbone colorings along matchings and stars

AU - Broersma, Haitze J.

AU - Marchal, L.

AU - Paulusma, Daniël

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N2 - 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.

AB - 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.

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BT - Proceedings of SOFSEM 2007: Theory and Practice of Computer Science

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A2 - Plásil, F.

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Broersma HJ, Marchal L, Paulusma D, Salman M. Improved upper bounds for λ-backbone colorings along matchings and stars. In van Leeuwen J, Italiano GF, van der Hoek W, Meinel C, Sack H, Plásil F, editors, Proceedings of SOFSEM 2007: Theory and Practice of Computer Science. Berlin: Springer. 2007. p. 188-199. 10.1007/978-3-540-69507-3. (Lecture Notes in Computer Science; LNCS4549). https://doi.org/10.1007/978-3-540-69507-3_15, https://doi.org/10.1007/978-3-540-69507-3