No abstract

No abstract

A town has a network of walkways running through it, each colored gray. A red bridge is constructed between every two walkway intersections that are two blocks apart, and a blue bridge is constructed between every two intersections that are three blocks apart. The colored bridges problem asks whether it is possible to take a round-trip about the town passing through each of its intersections exactly once such that during this trip the walkway or bridge used to enter every intersection is of a different color than that used to exit it. With the aid of graph theory, we show that such a round-trip is always possible when the walkways of the town form a grid. Conditions are determined under which no blue bridges are needed.

For a strong oriented graph D of order n and diameter d and an integerFor 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 if D k contains a properly arc-colored Hamiltonian cycle. The smallest positive integer k for which D k is Hamiltonian-colored is the Hamiltonian coloring exponent hce(D) of D. For each integer n ≥ 3, the Hamiltonian coloring exponent of the directed cycle C n of order n is determined whenever this number exists. It is shown for each integer k ≥ 2 that there exists a strong oriented graph D k such that hce(D k ) = k with the added property that every properly colored Hamiltonian cycle in the kth power of D k must use all k colors. It is shown for every positive integer p there exists a a connected graph G with two different strong orientations D and D ′ such that hce(D) − hce(D ′ ) ≥ p.

In a properly vertex-colored graph G, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P. If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring of G. The minimum number of colors needed in a vertex rainbow coloring of G is the vertex rainbow connection number vrc(G) of G. Thus if G is a connected graph of order n ≥ 2, then 2 ≤ vrc(G) ≤ n. We present characterizations of all connected graphs G of order n for which vrc(G) ∈ {2, n − 1, n} and study the relationship between vrc(G) and the chromatic number χ(G) of G. For a connected graph G of order n and size m, the number m − n + 1 is the cycle rank of G. Vertex rainbow connection numbers are determined for all connected graphs of cycle rank 0 or 1 and these numbers are investigated for connected graphs of cycle rank 2.