Displaying similar documents to “The size of minimum 3-trees: cases 0 and 1 mod 12”

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.

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)....

Graphs with 4-Rainbow Index 3 and n − 1

Xueliang Li, Ingo Schiermeyer, 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 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...

On Closed Modular Colorings of Trees

Bryan Phinezy, Ping Zhang (2013)

Discussiones Mathematicae Graph Theory

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Two vertices u and v in a nontrivial connected graph G are twins if u and v have the same neighbors in V (G) − {u, v}. If u and v are adjacent, they are referred to as true twins; while if u and v are nonadjacent, they are false twins. For a positive integer k, let c : V (G) → Zk be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex coloring c′ : V (G) → Zk defined by c′(v) = P u∈N[v] c(u) for each v ∈ V (G), where N[v] is...

Locally bounded -colorings of trees

C. Bentz, C. Picouleau (2009)

RAIRO - Operations Research

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Given a tree with vertices, we show, by using a dynamic programming approach, that the problem of finding a 3-coloring of respecting local (, associated with prespecified subsets of vertices) color bounds can be solved in log) time. We also show that our algorithm can be adapted to the case of -colorings for fixed .

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. ...