Abstract
The conditions under which both graph G and its complement ¯G share some common properties, e.g., with diameter 2, with ℓ 1-addressings, with property of being strongly regular are studied. Many examples and counterexamples from different areas of graph theory regarding them are provided. In particular, a census of graphs with at most six vertices is given.
Preview
Unable to display preview. Download preview PDF.
References
P. Assouad & M. Deza, Espaces metrique plongeables dans un hypercube: aspects combinatoires, Annals of Discrete Mathematics 8(1980) 197–210.
J. Akiyama & F. Harary, A graph and its complement with specified properties 1: connectivity, International Journal of Mathematics and Mathematical Sciences 2(1979) 223–228.
R. Alter, A characterization of self-complementary graphs of order 8, Portugaliae Mathematica 34(1975) 157–161.
A. E. Brouwer, A. M. Cohen, and A. Neumaier, “Distance-Regular Graphs”, Springer-Verlag, 1989.
J. Bosak, “Decompositions of Graphs”, Kluwer Academic Publishers, 1990.
J. Bosak, P. Erdös and A. Rosa, Decompositions of complete graphs into factors with diameter 2, Matematiickj Casopis 21(1971) 14–28.
J. Bosak, A. Rosa and S. Znam, On decomposition of complete graphs into factors with given diameters, in “Theory of Graphs”, P. Erdös, G. Katona (eds.), Academic Press, (1969) 37–56.
D. Cvetković & M. Peric, A table of connected graphs on six verticces, Discrete Mathematics 50(1984) 37–49.
P. K. Das, Almost self-complementary graphs, Ars Combinatoria 31(1991) 267–276.
M. Deza, On the Hamming geometry of unitary cubes, Doklady Akademii Nauk SSR (in Russian) 1037–1040, Soviet Physics Doklady 5(1961) 940–943.
M. Deza and T. Huang, ℓ 1-embeddability of some block graphs and cycloids, Bulletin of the Institute of Mathematics, Academia Sinica, to appear (1996).
M. Deza and V.P. Grishukhin, Hypermetric graphs, Quarterly Journal of Oxford (2) (1993), 399–433.
M. Deza and M. Laurent, ℓ 1-rigid graphs, Journal of Algebraic Combinatorics 3(1994), 153–175.
M. Deza and H. Maehara, Projecting a simplex onto another one, European Journal of Combinatorics 15(1994), 13–16.
R. C. Entringer, D. E. Jackson and P. A. Snyder, Distance in graphs, Czechoslovak Mathematical Journal 26(1976), 283–296.
R. A. Gibbs, Self-complementary graphs, Journal of Combinatorial Theory, Ser. B, 16(1974) 106–123.
R. L. Graham, B. L. Rothschild, and J. H. Spencer, “Ramsey Theory” 2nd. ed., John-Wiley & Sons (1990).
N. Hartsfield, On regular self-complementary graphs, Journal of Graph Theory 11(1987) 537–538.
D. Hanson, G. MacGillary, On small triangle-free graphs, Ars combinatoria 35(1993) 257–263.
R. Laskar, S. Stueckle and B. Piazza, On the edge-integrity of some graphs and their complements, Discrete Mathematics 122(1993) 245–253.
T. D. Parsons, Orthogonality graphs, Ars Combinatoria 3(1977), 165–208.
T. D. Parsons and T. Pisanki, Exotic n-universal graphs, Journal of Graph Theory 12(1988) 159–167.
I. G. Rosenberg, Regular and strongly regular self-complementary graphs, Annals of Discrete Mathematics 12(1982) 223–238.
S. Ruiz, On strongly regular self-complementary graphs, Journal of Graph Theory 5(1981), 213–215.
S. V. Shpectorov, On scale embeddings of graphs into hypercubes, European Journal of Combinatorics 14(1993) 117–130.
P. Terwilliger and M. Deza, Classification of finite connected hypermetric spaces, Graph and Combinatorics 3(1987), 293–298.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Deza, M., Huang, T. (1996). Complementary ℓ1-graphs and related combinatorial structures. In: Deza, M., Euler, R., Manoussakis, I. (eds) Combinatorics and Computer Science. CCS 1995. Lecture Notes in Computer Science, vol 1120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61576-8_76
Download citation
DOI: https://doi.org/10.1007/3-540-61576-8_76
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61576-7
Online ISBN: 978-3-540-70627-4
eBook Packages: Springer Book Archive