Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl
Preferencje help
Polish version English version Język
Widoczny [Schowaj] Abstrakt
Liczba wyników

Znaleziono wyników: 3

Liczba wyników na stronie
first rewind previous Strona / 1 next fast forward last

Wyniki wyszukiwania

help Sortuj według:

help Ogranicz wyniki do:
first rewind previous Strona / 1 next fast forward last
1
Content available remote

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

100%
Li X., Schiermeyer I., Yang K., Zhao Y.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 1
105-120
EN
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 rxk(G), where k is an integer such that 2 ≤ k ≤ n. Chartrand et al. got that the k-rainbow index of a tree is n−1 and the k-rainbow index of a unicyclic graph is n−1 or n−2. So there is an intriguing problem: Characterize graphs with the k-rainbow index n − 1 and n − 2. In this paper, we focus on k = 3, and characterize the graphs whose 3-rainbow index is n − 1 and n − 2, respectively.
2
Content available remote

Graphs with 4-Rainbow Index 3 and n − 1

100%
Li X., Schiermeyer I., Yang K., Zhao Y.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 2
387-398
EN
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). Notice that a lower bound and an upper bound of the k-rainbow index of a graph with order n is k − 1 and n − 1, respectively. Chartrand et al. got that the k-rainbow index of a tree with order n is n − 1 and the k-rainbow index of a unicyclic graph with order n is n − 1 or n − 2. Li and Sun raised the open problem of characterizing the graphs of order n with rxk(G) = n − 1 for k ≥ 3. In early papers we characterized the graphs of order n with 3-rainbow index 2 and n − 1. In this paper, we focus on k = 4, and characterize the graphs of order n with 4-rainbow index 3 and n − 1, respectively.
3
Content available remote

The 3-Rainbow Index of a Graph

100%
Chen L., Li X., Yang K., Zhao Y.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 1
81-94
EN
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.
first rewind previous Strona / 1 next fast forward last
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.