Hardness Results for Total Rainbow Connection of Graphs

Lily Chen; Bofeng Huo; Yingbin Ma

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Displaying similar documents to “Hardness Results for Total Rainbow Connection of Graphs”

Rainbow Connection Number of Dense Graphs

Xueliang Li, Mengmeng Liu, Ingo Schiermeyer (2013)

Discussiones Mathematicae Graph Theory

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An edge-colored graph G is rainbow connected, if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this paper we show that rc(G) ≤ 3 if |E(G)| ≥ [...] + 2, and rc(G) ≤ 4 if |E(G)| ≥ [...] + 3. These bounds are sharp.

Bounds for the rainbow connection number of graphs

Ingo Schiermeyer (2011)

Discussiones Mathematicae Graph Theory

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An edge-coloured graph G is rainbow-connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow-connected. In this paper we show some new bounds for the rainbow connection number of graphs depending on the minimum degree and other graph parameters. Moreover, we discuss sharpness of some of these bounds. ...

WORM Colorings of Planar Graphs

J. Czap, S. Jendrol’, J. Valiska (2017)

Discussiones Mathematicae Graph Theory

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Given three planar graphs F,H, and G, an (F,H)-WORM coloring of G is a vertex coloring such that no subgraph isomorphic to F is rainbow and no subgraph isomorphic to H is monochromatic. If G has at least one (F,H)-WORM coloring, then W−F,H(G) denotes the minimum number of colors in an (F,H)-WORM coloring of G. We show that (a) W−F,H(G) ≤ 2 if |V (F)| ≥ 3 and H contains a cycle, (b) W−F,H(G) ≤ 3 if |V (F)| ≥ 4 and H is a forest with Δ (H) ≥ 3, (c) W−F,H(G) ≤ 4 if |V (F)| ≥ 5 and H is...

Note on highly connected monochromatic subgraphs in 2-colored complete graphs.

Fujita, Shinya, Magnant, Colton (2011)

The Electronic Journal of Combinatorics [electronic only]

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Rainbow Connection Number of Graphs with Diameter 3

Hengzhe Li, Xueliang Li, Yuefang Sun (2017)

Discussiones Mathematicae Graph Theory

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A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer k for which there exists a k-edge-coloring of G such that every pair of distinct vertices of G is connected by a rainbow path. Let f(d) denote the minimum number such that rc(G) ≤ f(d) for each bridgeless graph G with diameter d. In this paper, we shall show that 7 ≤ f(3) ≤ 9.

Rainbow paths with prescribed ends.

Alishahi, Meysam, Taherkhani, Ali, Thomassen, Carsten (2011)

The Electronic Journal of Combinatorics [electronic only]

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Coloring with no 2-colored $P_{4}$ 's.

Albertson, Michael O., Chappell, Glenn G., Kierstead, H.A., Kündgen, André, Ramamurthi, Radhika (2004)

The Electronic Journal of Combinatorics [electronic only]

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Characterizations of Graphs Having Large Proper Connection Numbers

Chira Lumduanhom, Elliot Laforge, Ping Zhang (2016)

Discussiones Mathematicae Graph Theory

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Let G be an edge-colored connected graph. A path P is a proper path in G if no two adjacent edges of P are colored the same. If P is a proper u − v path of length d(u, v), then P is a proper u − v geodesic. An edge coloring c is a proper-path coloring of a connected graph G if every pair u, v of distinct vertices of G are connected by a proper u − v path in G, and c is a strong proper-path coloring if every two vertices u and v are connected by a proper u− v geodesic in G. The minimum...

Analogues of cliques for oriented coloring

William F. Klostermeyer, Gary MacGillivray (2004)

Discussiones Mathematicae Graph Theory

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We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.

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....

On rainbow connection.

Caro, Yair, Lev, Arie, Roditty, Yehuda, Tuza, Zsolt, Yuster, Raphael (2008)

The Electronic Journal of Combinatorics [electronic only]

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The set chromatic number of a graph

Gary Chartrand, Futaba Okamoto, Craig W. Rasmussen, Ping Zhang (2009)

Discussiones Mathematicae Graph Theory

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For a nontrivial connected graph G, let c: V(G)→ N be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G, the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) ≠ NC(v) for every pair u,v of adjacent vertices of G. The minimum number of colors required of such a coloring is called the set chromatic number χₛ(G) of G. The set chromatic numbers of some well-known classes of graphs...

Vertex rainbow colorings of graphs

Futaba Fujie-Okamoto, Kyle Kolasinski, Jianwei Lin, Ping Zhang (2012)

Discussiones Mathematicae Graph Theory

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In a properly vertex-colored graph G, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P. If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring of G. The minimum number of colors needed in a vertex rainbow coloring of G is the vertex rainbow connection number...

Rainbow Connection In Sparse Graphs

Arnfried Kemnitz, Jakub Przybyło, Ingo Schiermeyer, Mariusz Woźniak (2013)

Discussiones Mathematicae Graph Theory

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An edge-coloured connected graph G = (V,E) is called rainbow-connected if each pair of distinct vertices of G is connected by a path whose edges have distinct colours. The rainbow connection number of G, denoted by rc(G), is the minimum number of colours such that G is rainbow-connected. In this paper we prove that rc(G) ≤ k if |V (G)| = n and for all integers n and k with n − 6 ≤ k ≤ n − 3. We also show that this bound is tight.

Worm Colorings

Wayne Goddard, Kirsti Wash, Honghai Xu (2015)

Discussiones Mathematicae Graph Theory

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Given a coloring of the vertices, we say subgraph H is monochromatic if every vertex of H is assigned the same color, and rainbow if no pair of vertices of H are assigned the same color. Given a graph G and a graph F, we define an F-WORM coloring of G as a coloring of the vertices of G without a rainbow or monochromatic subgraph H isomorphic to F. We present some results on this concept especially as regards to the existence, complexity, and optimization within certain graph classes....

Precise Upper Bound for the Strong Edge Chromatic Number of Sparse Planar Graphs

Oleg V. Borodin, Anna O. Ivanova (2013)

Discussiones Mathematicae Graph Theory

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We prove that every planar graph with maximum degree ∆ is strong edge (2∆−1)-colorable if its girth is at least 40 [...] +1. The bound 2∆−1 is reached at any graph that has two adjacent vertices of degree ∆.

Bounds for the b-Chromatic Number of Subgraphs and Edge-Deleted Subgraphs

P. Francis, S. Francis Raj (2016)

Discussiones Mathematicae Graph Theory

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A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. In this paper, we obtain bounds for the b- chromatic number of induced subgraphs in terms of the b-chromatic number of the original graph. This turns out to be...