On conjectures of Frankl and El-Zahar

An induced subgraph ${\cal S}$ of a graph ${\cal G}$ is called a derived subgraph of ${\cal G}$ if ${\cal S}$ contains no isolated vertices. An edge e of ${\cal S}$ is said to be residual if e occurs in more than half of the derived subgraphs of ${\cal S}$. In this article, we prove that every simple graph with at least one edge contains a non-residual edge. This was conjectured by El-Zahar in 1997. © 2008 Wiley Periodicals, Inc. J Graph Theory 57: 344–352, 2008

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