Abstract
A matching in a graph is uniquely restricted if no other matching covers exactly the same set of vertices. This notion was defined by Golumbic, Hirst, and Lewenstein and studied in a number of articles. Our contribution is twofold. We provide approximation algorithms for computing a uniquely restricted matching of maximum size in some bipartite graphs. In particular, we achieve a ratio of 5/9 for subcubic bipartite graphs, improving over a 1/2-approximation algorithm proposed by Mishra. Furthermore, we study the uniquely restricted chromatic index of a graph, defined as the minimum number of uniquely restricted matchings into which its edge set can be partitioned. We provide tight upper bounds in terms of the maximum degree and characterize all extremal graphs. Our constructive proofs yield efficient algorithms to determine the corresponding edge colorings.
This work has been supported by the DE-MO-GRAPH grant ANR-16-CE40-0028.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alon, N., Sudakov, B., Zaks, A.: Acyclic edge colorings of graphs. J. Graph Theory 37, 157–167 (2001)
Andersen, L.: The strong chromatic index of a cubic graph is at most 10. Discret. Math. 108, 231–252 (1992)
Brandstädt, A., Mosca, R.: On distance-3 matchings and induced matchings. Discret. Appl. Math. 159, 509–520 (2011)
Bruhn, H., Joos, F.: A stronger bound for the strong chromatic index. arXiv:1504.02583 (2015)
Cai, X., Perarnau, G., Reed, B., Watts, A.: Acyclic edge colourings of graphs with large girth. arXiv:1411.3047 (2014)
Cameron, K.: Induced matchings. Discret. Appl. Math. 24, 97–102 (1989)
Cameron, K.: Induced matchings in intersection graphs. Discret. Math. 278, 1–9 (2004)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics. Springer, Heidelberg (2005)
Esperet, L., Parreau, A.: Acyclic edge-coloring using entropy compression. Eur. J. Comb. 34, 1019–1027 (2013)
Faudree, R., Gyárfás, A., Schelp, R., Tuza, Z.: Induced matchings in bipartite graphs. Discret. Math. 78, 83–87 (1989)
Faudree, R., Schelp, R., Gyárfás, A., Tuza, Z.: The strong chromatic index of graphs. Ars Combinatoria 29B, 205–211 (1990)
Fiamčik, J.: The acyclic chromatic class of a graph. Mathematica Slovaca 28, 139–145 (1978)
Francis, M.C., Jacob, D., Jana, S.: Uniquely restricted matchings in interval graphs. arXiv:1604.07016 (2016)
Golumbic, M., Hirst, T., Lewenstein, M.: Uniquely restricted matchings. Algorithmica 31, 139–154 (2001)
Henning, M., Rautenbach, D.: Induced matchings in subcubic graphs without short cycles. Discret. Math. 35–316, 165–172 (2014)
Hocquard, H., Montassier, M., Raspaud, A., Valicov, P.: On strong edge-colouring of subcubic graphs. Discret. Appl. Math. 161, 2467–2479 (2013)
Hocquard, H., Ochem, P., Valicov, P.: Strong edge-colouring and induced matchings. Inf. Process. Lett. 113, 836–843 (2013)
Horák, P., Qing, H., Trotter, W.: Induced matchings in cubic graphs. J. Graph Theory 17, 151–160 (1993)
Joos, F., Rautenbach, D., Sasse, T.: Induced matchings in subcubic graphs. SIAM J. Discret. Math. 28, 468–473 (2014)
Kang, R., Mnich, M., Müller, T.: Induced matchings in subcubic planar graphs. SIAM J. Discret. Math. 26, 1383–1411 (2012)
Levit, V., Mandrescu, E.: Very well-covered graphs of girth at least four and local maximum stable set greedoids. Discret. Math. Algorithms Appl. 3, 245–252 (2011)
Levit, V.E., Mandrescu, E.: Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings. Discret. Appl. Math. 132, 163–174 (2003)
Levit, V.E., Mandrescu, E.: Unicycle graphs and uniquely restricted maximum matchings. Electron. Notes Discret. Math. 22, 261–265 (2005)
Lovász, L.: Three short proofs in graph theory. J. Comb. Theory Ser. B 19, 269–271 (1975)
Lovász, L., Plummer, M.: Matching Theory. North-Holland, Amsterdam (1986)
Lozin, V.: On maximum induced matchings in bipartite graphs. Inf. Process. Lett. 81, 7–11 (2002)
Mishra, S.: On the maximum uniquely restricted matching for bipartite graphs. Electron. Notes Discret. Math. 37, 345–350 (2011)
Molloy, M., Reed, B.: A bound on the strong chromatic index of a graph. J. Comb. Theory Ser. B 69, 103–109 (1997)
Penso, L., Rautenbach, D., Souza, U.: Graphs in which some and every maximum matching is uniquely restricted. arXiv:1504.02250 (2015)
Stockmeyer, L., Vazirani, V.: NP-completeness of some generalizations of the maximum matching problem. Inf. Process. Lett. 15, 14–19 (1982)
Vizing, V.: On an estimate of the chromatic class of a p-graph. Diskretnyj Analiz 3, 25–30 (1964)
Yu, Q.R., Liu, G.: Graph Factors and Matching Extensions. Springer; Higher Education Press, Berlin; Beijing (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Baste, J., Rautenbach, D., Sau, I. (2017). Uniquely Restricted Matchings and Edge Colorings. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-68705-6_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68704-9
Online ISBN: 978-3-319-68705-6
eBook Packages: Computer ScienceComputer Science (R0)