A005488 - OEIS

%I M2528 #24 May 05 2022 08:08:41

%S 0,1,3,6,9,13,18,24,29

%N Maximal number of edges in a b^{hat} graceful graph with n nodes.

%C A graph with e edges is 'b^{hat} graceful' if its nodes can be labeled with distinct nonnegative integers so that, if each edge is labeled with the absolute difference between the labels of its endpoints, then the e edges have the distinct labels 1, 2, ..., e.

%C Equivalently, maximum m for which there's a difference basis with respect to m with n elements. A 'difference basis w.r.t. m' is a set of integers such that every integer from 1 to m is a difference between two elements of the set.

%C Miller's paper gives these lower bounds for the 11 terms from a(9) to a(19): 29,37,45,51,61,70,79,93,101,113,127. (Bermond's paper gives these as exact values, but quotes Miller as their source.)

%D J.-C. Bermond, Graceful graphs, radio antennae and French windmills, pp. 18-37 of R. J. Wilson, editor, Graph Theory and Combinatorics. Pitman, London, 1978.

%D R. K. Guy, Unsolved Problems in Number Theory, Sect. C10.

%D J. C. P. Miller, Difference bases: Three problems in additive number theory, pp. 299-322 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H J. Leech, <a href="https://doi.org/10.1112/jlms/s1-31.2.160">On the representation of 1, 2, ..., n by differences</a>, J. Lond. Math. Soc. 31 (1956), 160-169.

%e a(7)=18: Label the 7 nodes 0,6,9,10,17,22,24 and include all edges except those from 0 to 22, from 0 to 24 and from 17 to 24. {0,6,9,10,17,22,24} is a difference basis w.r.t. 18.

%Y Cf. A004137, A007187, A239308.

%K nonn,more

%O 1,3

%A _N. J. A. Sloane_, _Simon Plouffe_

%E Edited by _Dean Hickerson_, Jan 26 2003

%E a(9) from _J. Stauduhar_, May 04 2022