NDLI: On Additive Bases and Harmonious Graphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Graham, R. L. Sloane, N. J. A.
Copyright Year	1980
Abstract	This paper first considers several types of additive bases. A typical problem is to find $n_\gamma ( k )$, the largest n for which there exists a set $\{ 0 = a_1 < a_2 < \cdots < a_k \}$ of distinct integers modulo n such that each r in the range $0\leqq r \leqq n - 1$ can be written at least once as $r \equiv a_i + a_j $ (modulo n) with $i < j$. For example, $n_\gamma ( 8 ) = 24$, as illustrated by the set {0, 1, 2, 4, 8, 13, 18, 22}. The other problems arise if at least is changed to at most, or $i < j$ to $i\leqq j$, or if the words modulo n are omitted. Tables and bounds are given for each of these problems. Then a closely related graph labeling problem is studied. A connected graph with n edges is called harmonious if it is possible to label the vertices with distinct numbers (modulo n) in such a way that the edge sums are also distinct (modulo n). Some infinite families of graphs (odd cycles, ladders, wheels, $ \cdots $) are shown to be harmonious while others (even cycles, most complete or complete bipartite graphs, $ \cdots $) are not. In fact most graphs are not harmonious. The function $n_\gamma ( k )$ is the size of the largest harmonious subgraph of the complete graph on k vertices.
Starting Page	382
Ending Page	404
Page Count	23
File Format	PDF
ISSN	01965212
DOI	10.1137/0601045
Journal	SIAM Journal on Algebraic Discrete Methods (SJAMDU)
Issue Number	4
Volume Number	1
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-08-02
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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