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Hamiltonian-colored powers of strong digraphs

Garry JohnsRyan JonesKyle KolasinskiPing Zhang — 2012

Discussiones Mathematicae Graph Theory

For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power D k of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of D k if the directed distance d D ( u , v ) from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph D k is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph D k is distance-colored if each arc (u, v) of D k is assigned the color i where i = d D ( u , v ) . The digraph D k is Hamiltonian-colored...

Rainbow connection in graphs

Gary ChartrandGarry L. JohnsKathleen A. McKeonPing Zhang — 2008

Mathematica Bohemica

Let G be a nontrivial connected graph on which is defined a coloring c E ( G ) { 1 , 2 , ... , k } , k , of the edges of G , where adjacent edges may be colored the same. A path P in G is a rainbow path if no two edges of P are colored the same. The graph G is rainbow-connected if G contains a rainbow u - v path for every two vertices u and v of G . The minimum k for which there exists such a k -edge coloring is the rainbow connection number r c ( G ) of G . If for every pair u , v of distinct vertices, G contains a rainbow u - v geodesic, then G is...

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