Abstract
We prove that the edge set of an arbitrary simple graphG onn vertices can be covered by at mostn−[log2 n]+1 complete bipartite subgraphs ofG. If the weight of a subgraph is the number of its vertices, then there always exists a cover with total weightc(n 2/logn) and this bound is sharp apart from a constant factor. Our result answers a problem of T. G. Tarján.
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References
J. C. Bermond, Couverture des arrêtes d’un graphe par des graphes bipartis complets,preprint, Univ. de Paris-Sud, Centre d’Orsay, Rapport de Recherche No. 10. (June 1978).
B. Bollobás, On complete subgraphs of different orders,Math. Proc. Comb. Philos. Soc. 79 (1976), 19–24.
N. G. DeBruijn andP. Erdős. On a combinatorial problem,Nederl. Akad. Wetensch. Proc. 51 (1948), 1277–1279.
F. R. K. Chung, On the coverings of graphs,Discrete Math. 30 (1980), 89–93.
P. Erdős, Graph theory and probability,Canad. J. Math. 11 (1959) 34–38.
P. Erdős, A. Goodman andL. Pósa, The representation of graphs by set intersections,Canad. J. Math. 18 (1966), 106–112.
P. Erdős andG. Szekeres, A combinatorial problem in geometry,Compositio Math. 2 (1935), 463–470.
R. L. Graham andL. Lovász, Distance matrix polynomials of trees,Advances in Mathematics 29 (1978), 60–88.
E. Győri andA. V. Kostochka, On a problem of G. O. H. Katona and T. Tarján,Acta Math. Acad. Sci. Hungar. 34 (1979), 321–327.
F. Harary, D. Hsu andZ. Miller, The biparticity of a graph,J. Graph Theory 1 (1977), 131–133.
C. Hylten-Cavallius, On a combinatorial problem,Colloq. Math. 6 (1958), 59–65.
G. Katona andE. Szemerédi, On a problem of graph theory,Studia Sci. Math. Hungar. 2 (1967), 23–28.
J. Lehel, Covers in hypergraphs,Combinatorica 2 (3) (1982), 305–309.
J. Lehel andZs. Tuza, Triangle-free partial graphs and edge covering theorems,Discrete Math. 39 (1982), 59–65.
L. Lovász, On covering of graphs, in:Theory of Graphs, (P. Erdős and G. Katona, eds.),Proc. Coll. at Tihany, Hungary, 1966, Academic Press and Akadémiai Kiadó, 231–236.
V. Rödl, private communication.
T. G. Tarján, Complexity of lattice-configurations,Studia Sci. Math. Hungar. 10 (1975), 203–211.
H. Tverberg, On the decomposition ofK n in complete bipartite graphs,J. Graph Theory 6 (1982), 493–494.
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Dedicated to Paul Erdős on his seventieth birthday