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Graphs S ( n , k ) and a variant of the Tower of Hanoi problem

Sandi Klavžar, Uroš Milutinović (1997)

Czechoslovak Mathematical Journal

For any n ≥ 1 and any k ≥ 1 , a graph S ( n , k ) is introduced. Vertices of S ( n , k ) are n -tuples over { 1 , 2 , ... , k } and two n -tuples are adjacent if they are in a certain relation. These graphs are graphs of a particular variant of the Tower of Hanoi problem. Namely, the graphs S ( n , 3 ) are isomorphic to the graphs of the Tower of Hanoi problem. It is proved that there are at most two shortest paths between any two vertices of S ( n , k ) . Together with a formula for the distance, this result is used to compute the distance between two vertices in...

Graphs with 3-Rainbow Index n − 1 and n − 2

Xueliang Li, Ingo Schiermeyer, Kang Yang, Yan Zhao (2015)

Discussiones Mathematicae Graph Theory

Let G = (V (G),E(G)) be a nontrivial connected graph of order n with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ N, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree connecting S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbow S-tree for each k-subset S of V (G) is called the k-rainbow index of G, denoted by...

Graphs with 4-Rainbow Index 3 and n − 1

Xueliang Li, Ingo Schiermeyer, Kang Yang, Yan Zhao (2015)

Discussiones Mathematicae Graph Theory

Let G be a nontrivial connected graph with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ ℕ, where adjacent edges may be colored the same. A tree T in G is called a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree that connects S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbow S-tree for every set S of k vertices of V (G) is called the k-rainbow index of G, denoted by rxk(G)....

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