Automorphic Decompositions of Graphs

A decomposition $${\fancyscript {D}}$$ of a graph H by a graph G is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. The intersection graph $${I(\fancyscript {D})}$$ of the decomposition $${\fancyscript {D}}$$ has a vertex for each part of the partition and two parts A and B are adjacent iff they share a common node in H. If $${I(\fancyscript {D})\cong H}$$, then $${\fancyscript {D}}$$ is an automorphic decomposition of H. In this paper we show how automorphic decompositions serve as a common generalization of configurations from geometry and graceful labelings on graphs. We will give several examples of automorphic decompositions as well as necessary conditions for their existence.