# Graph Posets, Spanning-Universal Graphs and Graph Coverings

MacDougall, James A. and Eggleton, Roger B. (2003) Graph Posets, Spanning-Universal Graphs and Graph Coverings. Bulletin in the I.C.A., 37 . pp. 63-72.

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The poset of simple graphs of order $n$ is a lattice if and only if $n \leq 4$, whereas the poset of labelled simple graphs of order $n$ is always a lattice. For any family $S$ of simple graphs of order $n$, we use the ideals in these posets to conveniently define the spanning-universal graphs for $S$ and the graphs heterogeneously covered by $S$. In a related manner we define the graphs homogeneously covered by any simple graph $H$ of order $n$. We describe a number of concrete cases for small n when $S \subseteq T(n)$ or $H \in T(n)$ where $T(n)$ is the set of trees of order $n$, and we pose several general problems.