ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 22(1) (2002) 89-99
DOI: 10.7151/dmgt.1159


Jirí Fiala and  Jan Kratochvíl

Department of Applied Mathematics and Institute
for Theoretical Computer Science, Charles University
Malostranské nám. 25, 118 00 Prague, Czech Republic


Given graphs G and H, a mapping f:V(G)→ V(H) is a homomorphism if (f(u),f(v)) is an edge of H for every edge (u,v) of G. In this paper, we initiate the study of computational complexity of locally injective homomorphisms called partial covers of graphs. We motivate the study of partial covers by showing a correspondence to generalized (2,1)-colorings of graphs, the notion stemming from a practical problem of assigning frequencies to transmitters without interference. We compare the problems of deciding existence of partial covers and of full covers (locally bijective homomorphisms), which were previously studied.

Keywords: covering projection, computational complexity, graph homomorphism

2000 Mathematics Subject Classification: 05C85, 05C78.


[1] J. Abello, M.R. Fellows and J.C. Stillwell, On the complexity and combinatorics of covering finite complexes, Australian Journal of Combinatorics 4 (1991) 103-112.
[2] D. Angluin, Local and global properties in networks of processors, in: Proceedings of the 12th ACM Symposium on Theory of Computing (1980) 82-93.
[3] H.L. Bodlaender, The classification of coverings of processor networks, Journal of Parallel Distributed Computing 6 (1989) 166-182, doi: 10.1016/0743-7315(89)90048-8.
[4] N. Biggs, Algebraic Graph Theory (Cambridge University Press, 1974).
[5] G.J. Chang and D. Kuo, The L(2,1)-labeling problem on graphs, SIAM J. Disc. Math. 9 (1996) 309-316, doi: 10.1137/S0895480193245339.
[6] B. Courcelle and Y. Métivier, Coverings and minors: Applications to local computations in graphs, European Journal of Combinatorics 15 (1994) 127-138, doi: 10.1006/eujc.1994.1015.
[7] M. Dyer and C. Greenhill, The complexity of counting graph homomorphisms, in: Proceedings of the Eleventh annual ACM-SIAM Symposium on Discrete Algorithms SODA'00 (San Francisco, 2000).
[8] J. Fiala, Locally injective homomorphism (Doctoral Thesis, Charles University, Praque 2000).
[9] J. Fiala and J. Kratochvíl, Complexity of partial covers of graphs, in: Algorithms and Computation, 12th ISAAC'01, Christchurch, New Zealand, Lecture Notes in Computer Science 2223 (Springer Verlag, 2001) 537-549.
[10] J. Fiala, J. Kratochvíl and T. Kloks, Fixed-parameter complexity of λ-labelings, Discrete Applied Math. 113 (1) (2001) 59-72, doi: 10.1016/S0166-218X(00)00387-5.
[11] J.R. Griggs and R.K. Yeh, Labelling graphs with a condition at distance 2, SIAM J. Disc. Math. 5 (1992) 586-595, doi: 10.1137/0405048.
[12] J.L. Gross and T.W. Tucker, Topological Graph Theory (J. Wiley and Sons, 1987).
[13] J.L. Gross and T.W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Math. 18 (1977) 273-283, doi: 10.1016/0012-365X(77)90131-5.
[14] P. Hell and J. Nesetril, On the complexity of H-coloring, J. Combin. Theory (B) 48 (1990) 92-110, doi: 10.1016/0095-8956(90)90132-J.
[15] P. Hell and J. Nesetril, The core of a graph, Discrete Math. 109 (1992) 117-126, doi: 10.1016/0012-365X(92)90282-K.
[16] J. van den Heuvel, R.A. Leese and M.A. Shepherd, Graph labeling and radio channel assignment, Journal of Graph Theory 29 (4) (1998) 263-283, doi: 10.1002/(SICI)1097-0118(199812)29:4<263::AID-JGT5>3.0.CO;2-V.
[17] J. Kratochvíl, A. Proskurowski and J.A. Telle, Covering regular graphs, J. Combin. Theory (B) 71 (1997) 1-16, doi: 10.1006/jctb.1996.1743.
[18] J. Kratochvíl, A. Proskurowski and J.A. Telle, Complexity of graph covering problems, Nordic Journal of Computing 5 (1998) 173-195.
[19] J. Kratochvíl, A. Proskurowski and J.A. Telle, Covering directed multigraphs I. Colored directed multigraphs, in: Graph-Theoretical Concepts in Computer Science, Proceedings 23rd WG '97, Berlin, Lecture Notes in Computer Science 1335 (Springer Verlag, 1997) 242-254.
[20] R.A. Leese, A fresh look at channel assignment in uniform networks, EMC97 Symposium, Zurich, 127-130.
[21] F.T. Leighton, Finite common coverings of graphs, J. Combin. Theory (B) 33 (1982) 231-238, doi: 10.1016/0095-8956(82)90042-9.
[22] W. Massey, Algebraic Topology: An Introduction (Harcourt, Brace and World, 1967).
[23] K.-Ch. Yeh, Labeling graphs with a condition at distance two, Ph.D. Thesis (University of South Carolina, 1990).

Received 7 August 2000
Revised 18 June 2001