Sets of lattice points which contain a maximal number of edges
Author:
G. F. Clements
Journal:
Proc. Amer. Math. Soc. 27 (1971), 13-15
MSC:
Primary 05.04
DOI:
https://doi.org/10.1090/S0002-9939-1971-0270923-7
MathSciNet review:
0270923
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: How should one select an -element subset of a rectangular array of lattice points (points with integral coordinates) in
-dimensional Euclidean space so as to include the largest possible number of edges (pairs of points differing in exactly one coordinate)? It is shown that the generalized Macaulay theorem due to the author and B. Lindström contains the (known) solution.
- [1]
A. J. Bernstein, Maximally connected arrays on the
-cube, SIAM J. Appl. Math. 15 (1967), 1485-1489. MR 36 #6308. MR 0223260 (36:6308)
- [2] G. F. Clements and B. Lindström, A generalization of a combinatorial theorem of Macaulay, J. Combinatorial Theory 7 (1969), 230-238. MR 0246781 (40:50)
- [3] L. H. Harper, Optimal assignments of numbers to vertices, J. Soc. Indust. Appl. Math. 12 (1964), 131-135. MR 29 #41. MR 0162737 (29:41)
- [4] G. Katona, A theorem of finite sets, Proc. Colloq. Theory of Graphs (Tihany, Hungary, 1966) Academic Press, New York and Akad. Kiadó, Budapest, 1968. MR 38 #1016. MR 0290982 (45:76)
- [5] J. B. Kruskal, The number of simplices in a complex, Mathematical Optimization Techniques, Univ. of California Press, Berkeley, Calif., 1963, pp. 251-278. MR 27 #4771. MR 0154827 (27:4771)
- [6]
-, The number of
-dimensional faces in a complex: An analogy between the simplex and the cube, J. Combinatorial Theory 6 (1969), 86-89. MR 38 #4328. MR 0236030 (38:4328)
- [7] J. H. Lindsey, Assignment of numbers to vertices. Amer. Math. Monthly 71 (1964), 508-516. MR 29 #5751. MR 0168489 (29:5751)
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 05.04
Retrieve articles in all journals with MSC: 05.04
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1971-0270923-7
Keywords:
Lattice points,
maximal number of edges,
generalized Macaulay theorem,
-dimensional Euclidean space
Article copyright:
© Copyright 1971
American Mathematical Society