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The 3-Rainbow Index of a Graph

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Chen L., Li X., Yang K., Zhao Y.

Discussiones Mathematicae Graph Theory

2015

tom 35

nr 1

81-94

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 a rainbow tree if no two edges of T receive the same color. For a vertex subset 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 each k-subset S of V (G) is called the k-rainbow index of G, denoted by rxk(G). In this paper, we first determine the graphs of size m whose 3-rainbow index equals m, m − 1, m − 2 or 2. We also obtain the exact values of rx3(G) when G is a regular multipartite complete graph or a wheel. Finally, we give a sharp upper bound for rx3(G) when G is 2-connected and 2-edge connected. Graphs G for which rx3(G) attains this upper bound are determined.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Chen L.

1 Li X.

1 Yang K.

1 Zhao Y.

1 2015

Wyniki wyszukiwania

The 3-Rainbow Index of a Graph