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Vietnam Journal of Mathematics 34:3(2006) 285-294  Solution to an Open Problem on the Integral Sum Graphs Shuchao Li, Huiling Zhou, and Yanqin Feng Abstract.  The concept of the (integral) sum graphs was first introduced by Harary (Congr. Number 72 (1990) 101; Discrete Math. 124 (1994) 99). Let $N*$ denote the set of positive integers. The (integral) sum graph $G^+(S)$ of a finite subset $S\subset N*(Z)$ is the graph $(S,E)$ with $uv\in S$ if and only if $u+v\in S$. A graph $G$ is called an (integral) sum graph if it is isomorphic to the (integral) sum graph $G^+(S)$ for some $S\subset N^*(Z)$. In this paper we give a constructive method to show that the odd cycles are regular integral sum graphs, which extends the classes of integral sum graphs and completely solves an open problem posed by Baogen Xu (Discrete Math. 194 (1999) 285-294).   2000 Mathematics Subject Classification: 05C38, 05C78. Keywords: Odd cycle, integral sum graph.

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