Published online by Cambridge University Press: 20 November 2018
The annihilating-ideal graph of a commutative ring $R$ , denoted by
$\mathbb{A}\mathbb{G}\left( R \right)$ , is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices
$I$ and
$J$ are adjacent if and only if
$IJ\,=\,\left( 0 \right)$ . Here we show that if
$R$ is a reduced ring and the independence number of
$\mathbb{A}\mathbb{G}\left( R \right)$ is finite, then the edge chromatic number of
$\mathbb{A}\mathbb{G}\left( R \right)$ equals its maximum degree and this number equals
${{2}^{\left| \min \left( R \right) \right|-1}}-\,1$ ; also, it is proved that the independence number of
$\mathbb{A}\mathbb{G}\left( R \right)$ equals
${{2}^{\left| \min \left( R \right) \right|-1}}$ , where
$\min \left( R \right)$ denotes the set of minimal prime ideals of
$R$ . Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a finite graph
$\mathbb{A}\mathbb{G}\left( R \right)$ is not Eulerian, and that it is Hamiltonian if and only if
$R$ contains no Gorenstain ring as its direct summand.