A class of solutions to the gossip problem, part III

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Abstract

We complete the study of NOHO-graphs, begun in Parts I and II of this paper. NOHO- graphs correspond to solutions to the gossip problem where No One Hears his Own information. These are graphs with a linear ordering on their edges such that an increasing path exists from each vertex to every other, but from no vertex to itself. We discard the two such graphs with no 2-valent vertices. In Part I, we translated these graphs into quadruples of integer sequences. In Part II, we characterized and enumerated the realizable quadruples and various subclasses of them. In Part III, we eliminate the overcounting of isomorphic graphs and obtain recurrence relations and generating functions to enumerate the non-isomorphic NOHO-graphs. If um=(1,1,2,...) satisfies um=3um-1-um-3, then the number of non-isomorphic NOHO- graphs on 2m+2 vertices is 1 2(um + u[m/2]+1 + u[m/2]+1 - u[m/2]). We also examine some re lated questions. © 1982.

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West, D. B. (1982). A class of solutions to the gossip problem, part III. Discrete Mathematics, 40(2–3), 285–310. https://doi.org/10.1016/0012-365X(82)90128-5

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