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Optimal Graph Orientation with Storage Applications
, 1995
"... We show that the edges of a graph with maximum edge density d can always be oriented such that each vertex has in-degree at most d. Hence, for arbitrary graphs, edges can always be assigned to incident vertices as uniformly as possible. For example, in-degree 3 is achieved for planar graphs. This im ..."
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We show that the edges of a graph with maximum edge density d can always be oriented such that each vertex has in-degree at most d. Hence, for arbitrary graphs, edges can always be assigned to incident vertices as uniformly as possible. For example, in-degree 3 is achieved for planar graphs. This immediately gives a space-optimal data structure that answers edge membership queries in a maximum edge density-d graph in O(log d) time. Keywords Graph orientation, edge density, Hall condition, balanced adjacency lists, edge membership queries 1 The Theorem Let G be an undirected graph with n vertices and m edges. The parameter ffi(G) = m n is commonly called the edge density of G. The maximum (edge) density is the smallest integer d such that the edge density of any subgraph of G does not exceed d. More precisely, d = dmaxfffi(G 0 ) j G 0 is a subgraph of Gge. For example, d 1 for trees, d 3 for planar graphs, d = d 1 2 log 2 ne for hypercubes [GG,AH], and d d 1 2 (c \Gamma...
