Abstract
An injective coloring of a graph is a vertex labeling such that two vertices sharing a common neighbor get different labels. In this work we introduce and study what we call additive colorings. An injective coloring \(c:V(G)\rightarrow \mathbb {Z}\) of a graph \(G\) is an additive coloring if for every \(uv, vw\) in \(E(G)\), \(c(u)+c(w)\ne 2c(v)\). The smallest integer \(k\) such that an injective (resp. additive) coloring of a given graph \(G\) exists with \(k\) colors (resp. colors in \(\{1,\ldots ,k\}\)) is called the injective (resp. additive) chromatic number (resp. index). They are denoted by \(\chi _i(G)\) and \(\chi '_a(G)\), respectively. In the first part of this work, we present several upper bounds for the additive chromatic index. On the one hand, we prove a super linear upper bound in terms of the injective chromatic number for arbitrary graphs, as well as a linear upper bound for bipartite graphs and trees. Complete graphs are extremal graphs for the super linear bound, while complete balanced bipartite graphs are extremal graphs for the linear bound. On the other hand, we prove a quadratic upper bound in terms of the maximum degree. In the second part, we study the computational complexity of computing \(\chi '_a(G)\). We prove that it can be computed in polynomial time for trees. We also prove that for bounded treewidth graphs, to decide whether \(\chi '_a(G)\le k\), for a fixed \(k\), can be done in polynomial time. On the other hand, we show that for cubic graphs it is NP-complete to decide whether \(\chi '_a(G)\le 4\). We also prove that for every \(\epsilon >0\) there is a polynomial time approximation algorithm with approximation factor \(n^{1/3+\epsilon }\) for \(\chi '_a(G)\), when restricted to split graphs. However, unless \(\mathsf P =\mathsf NP \), for every \(\epsilon >0\) there is no polynomial time approximation algorithm with approximation factor \(n^{1/3-\epsilon }\) for \(\chi '_a(G)\), even when restricted to split graphs.
Similar content being viewed by others
Notes
In fact, they studied the dependency of \(m\) in terms of \(l\), which they called the Szemerédi number \(sz(l)\).
References
Behrend, F.A.: On the sets of integers which contain no three in arithmetic progression. Proc. Natl. Acad. Sci. 23, 331–332 (1946)
Bourgain, J.: On triples in arithmetic progression. Geom. Funct. Anal. 9, 968–984 (1999)
Brooks, R.L.: On colouring the nodes of a network. Proc. Camb. Phil. Soc. Math. Phys. Sci. 37, 194–197 (1941)
Chang, G.J., Kuo, D.: The \(L(2, 1)\)-labeling problem on graphs. SIAM J. Discrete Math. 9(2), 309–316 (1996)
Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)
Elkin, M.: An improved construction of progession-free sets. In: SODA: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 886–905 (2010)
Erdös, P., Turán, P.: On some sequences of integers. J. Lond. Math. Soc. 11, 261–264 (1936)
Feige, U., Killian, J.: Zero knowledge and the chromatic number. J. Comput. Syst. Sci. 57, 187–199 (1998)
Fiala, J., Golovach, P.A., Kratochvíl, J.: Computational complexity of the distance constrained labeling problem for trees (extended abstract). ICALP 1, 294–305 (2008)
Gasarch, W., Glenn, J., Kruskal, C.: Finding large 3-free sets. I. The small \(n\) case. J. Comput. System Sci. 74(4), 628–655 (2008)
Griggs, J.R., Yeh, R.K.: Labeling graphs with a condition at distance 2. SIAM J. Disc. Math. 5, 586–595 (1992)
Hahn, G., Kratochvîl, J., Širáň, J., Sotteau, D.: On the injective chromatic number of graphs. Discrete Math. 256(1–2), 179–192 (2002)
Heath-Brown, D.: Integer sets containing no arithmetic progressions. Proc. Lond. Math Soc. 35(2), 385–394 (1987)
Hell, P., Raspaud, A., Stacho, J.: On injective colourings of chordal graphs. In: LATIN: Proceeding Theoretical Informatics, 8th Latin American Symposium, pp. 520–530 (2008)
Holyer, I.: The NP-completeness of edge-coloring. SIAM J. Comput. 10(4), 718–720 (1981)
Hasunuma, T., Ishii, T., Ono, H., Uno, Y.: A linear time algorithm for \(L(2, 1)\)-labeling of trees. ESA, pp. 35–46 (2009)
Roth, K.: Sur quelques ensembles d’ entiers. C.R. Acad. Sci Paris 234, 388–390 (1952)
Roth, K.: On certain sets of integers. J. Lond. Math. Soc. 28, 104–109 (1953)
Singer, J.: A theorem of finite projective geometry and some applications to number theory. Trans. Am. Math. Soc. 43, 377–385 (1938)
Szemerédi, E.: Integer sets containing no arithmetic progressions. Acta Math. Sci. Hung. 56, 155–158 (1990)
Zuckermann, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: STOC: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 681–690, ACM, New York (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by Fondecyt 1100192, Ecos-Conicyt C09E04, Basal Program PBF 03, Beca de Doctorado Conicyt, Nucleo Milenio Información y Coordinación en Redes ICM/FIC P10-024F, French National Research Agency project AGAPE (ANR-09-BLAN-0159-03).
Rights and permissions
About this article
Cite this article
Astromujoff, N., Chapelle, M., Matamala, M. et al. Injective Colorings with Arithmetic Constraints. Graphs and Combinatorics 31, 2003–2017 (2015). https://doi.org/10.1007/s00373-014-1520-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-014-1520-3