OFFSET
1,3
COMMENTS
Comments from Don Knuth, Apr 17 2021: (Start)
Peter Winkler (1984) found a very nice way to decide whether a given graph can be isometrically embedded into a Hamming graph. He discovered that such an embedding is actually unique, when it exists.
Sequence A343463 and A343464 are about graphs that are embeddable as INDUCED subgraphs. The two concepts are quite distinct. Today I may have discovered for the first time the (unique) smallest graph that proves the distinction. This is the six-node graph:
* . -- .
* / | | \
* . | | .
* \ | | /
* . -- .
This graph cannot be isometrically embedded in a Hamming graph. But it is an induced subgraph:
* 10 -- 11
* / | | \
* 00 | | 31
* \ | | /
* 20 -- 21
Sequence A343168 is a triangular array that gives the number of graphs on n vertices that are isometrically embeddable in the Hamming graph H(n,k). The present sequence gives the row sums of this triangle, and is a direct analog of A343463. The present sequence and A343463 agree up to n=5, and the graph above is the first difference between them. (End)
REFERENCES
Donald E. Knuth, The Art of Computer Programming, Volume 4B
(in preparation). Section 7.2.2.3 will contain an exercise
devoted to this topic.
LINKS
Peter M. Winkler, Isometric embedding in products of complete graphs, Discrete Applied Mathematics 7 (1984), 221-225.
FORMULA
a(n) = Sum_{k=2..n} A343168(n, k). - Felix Fröhlich, Apr 22 2021
CROSSREFS
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, Apr 19 2021, based on an email from Don Knuth
STATUS
approved