A Note on the Road-Coloring Conjecture

13 years 4 months ago

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Some results relating to the road-coloring conjecture of Alder, Goodwyn, and Weiss, which give rise to an O(n2) algorithm to determine whether or not a given edge-coloring of a graph is a road-coloring, are noted. Probabilistic analysis is then used to show that, if the outdegree of every edge in an n-vertex digraph is δ = ω(log n), a road-coloring for the graph exists. An equivalent re-statement of the conjecture is then given in terms of the cross-product of two graphs. Definitions Let G be an n-vertex digraph. V (G) will denote the vertex-set of G, and E(G) will denote the edge-set of G. G is strongly connected if for every pair of vertices v and w in V (G), there is a directed path from v to w. The outdegree of vertex v ∈ V (G), d+ (v), is the number of edges originating at v. G is aperiodic if the set of lengths of simple directed cycles in G has gcd

E. Gocka, Walter W. Kirchherr, Edward F. Schmeiche

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