Embeddings of Complete Binary Trees into Grids and Extended Grids with Total Vertex-congestion 1

Abstract Let G and H be two simple undirected graphs An embedding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H together with a mapping which assigns to each edge u v of G a path between f u and f v in H The grid M r s is the graph whose vertex set is the set of pairs on nonnegative integers f i j i r j sg in which there is an edge between vertices i j and k l if either ji kj and j l or i k and jj lj The extended grid EM r s is the graph whose vertex set is the set of pairs on non negative integers f i j i r j sg in which there is an edge between vertices i j and k l if and only if ji kj and jj lj In this paper we give embeddings of complete binary trees into square grids and ex tended grids with total vertex congestion i e for any vertex x of the extended grid we have load x vertex congestion x Depending on the parity of the height of the tree the expansion of these embeddings is approaching or for grids and or for extended grids