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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 indegree at most d. Hence, for arbitrary graphs, edges can always be assigned to incident vertices as uniformly as possible. For example, indegree 3 is achieved for planar graphs. This im ..."
Abstract

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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 indegree at most d. Hence, for arbitrary graphs, edges can always be assigned to incident vertices as uniformly as possible. For example, indegree 3 is achieved for planar graphs. This immediately gives a spaceoptimal data structure that answers edge membership queries in a maximum edge densityd 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...