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Displaying similar documents to “Graphs with 4-Rainbow Index 3 and n − 1”

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

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

Discussiones Mathematicae Graph Theory

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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...

The 3-Rainbow Index of a Graph

Lily Chen, Xueliang Li, Kang Yang, Yan Zhao (2015)

Discussiones Mathematicae Graph Theory

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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)....

Upper Bounds for the Strong Chromatic Index of Halin Graphs

Ziyu Hu, Ko-Wei Lih, Daphne Der-Fen Liu (2018)

Discussiones Mathematicae Graph Theory

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The strong chromatic index of a graph G, denoted by χ′s(G), is the minimum number of vertex induced matchings needed to partition the edge set of G. Let T be a tree without vertices of degree 2 and have at least one vertex of degree greater than 2. We construct a Halin graph G by drawing T on the plane and then drawing a cycle C connecting all its leaves in such a way that C forms the boundary of the unbounded face. We call T the characteristic tree of G. Let G denote a Halin graph with...

Packing Trees Into n-Chromatic Graphs

András Gyárfás (2014)

Discussiones Mathematicae Graph Theory

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We show that if a sequence of trees T1, T2, ..., Tn−1 can be packed into Kn then they can be also packed into any n-chromatic graph.

Spanning trees with many or few colors in edge-colored graphs

Hajo Broersma, Xueliang Li (1997)

Discussiones Mathematicae Graph Theory

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Given a graph G = (V,E) and a (not necessarily proper) edge-coloring of G, we consider the complexity of finding a spanning tree of G with as many different colors as possible, and of finding one with as few different colors as possible. We show that the first problem is equivalent to finding a common independent set of maximum cardinality in two matroids, implying that there is a polynomial algorithm. We use the minimum dominating set problem to show that the second problem is NP-hard. ...

Parity vertex colorings of binomial trees

Petr Gregor, Riste Škrekovski (2012)

Discussiones Mathematicae Graph Theory

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We show for every k ≥ 1 that the binomial tree of order 3k has a vertex-coloring with 2k+1 colors such that every path contains some color odd number of times. This disproves a conjecture from [1] asserting that for every tree T the minimal number of colors in a such coloring of T is at least the vertex ranking number of T minus one.

Backbone colorings along stars and matchings in split graphs: their span is close to the chromatic number

Hajo Broersma, Bert Marchal, Daniel Paulusma, A.N.M. Salman (2009)

Discussiones Mathematicae Graph Theory

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We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph G = (V,E) and a spanning subgraph H of G (the backbone of G), a λ-backbone coloring for G and H is a proper vertex coloring V→ {1,2,...} of G in which the colors assigned to adjacent vertices in H differ by at least λ. The algorithmic and combinatorial properties of backbone colorings have been studied for various types of backbones in a number of papers....

Graphs with Large Generalized (Edge-)Connectivity

Xueliang Li, Yaping Mao (2016)

Discussiones Mathematicae Graph Theory

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The generalized k-connectivity κk(G) of a graph G, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized k-edge-connectivity λk(G). In this paper, graphs of order n such that [...] for even k are characterized.