Geometric Achromatic and Pseudoachromatic Indices

Aichholzer, Oswin; Araujo‐Pardo, Gabriela; García-Colín, Natalia; Hackl, Thomas; Lara, Dolores; Rubio-Montiel, Christian; Urrutia, Jorge

doi:10.1007/s00373-015-1610-x

Cited by 8 publications

(9 citation statements)

References 15 publications

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“…This distinguishes a crossing from an intersection of two edges. 15 The definition as given also distinguishes a crossing from the point in the plane at which the crossing occurs (and this is good). The definition does, however, include points in which two curves touch; this is of no consequence for the standard crossing number since in crossing-minimal drawings no touching points occur, but for other variants, e.g.…”

Section: A Systematic Approachmentioning

confidence: 99%

“…Harborth mentions an unpublished paper that seems to establish significant parts of this conjecture 15. One subtlety already: it excludes from the notion of crossing any intersection occurring when an edge passes through a vertex, as opposed to ending there.…”

mentioning

confidence: 99%

“…This is beautifully illustrated by an example from Bertin [80, Figure 4, p.109] 23. We should mention a paper[15], that repeatedly uses the term m + cr to denote the total number of crossings in a geometric drawing including m crossings of edges through vertices.the electronic journal of combinatorics (2020), #DS21…”

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confidence: 99%

See 2 more Smart Citations

The Graph Crossing Number and its Variants: A Survey

Schaefer

2013

Electron. J. Combin.

100

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The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number is the best known. We survey the rich variety of crossing number variants that have been introduced in the literature for purposes that range from studying the theoretical underpinnings of the crossing number to crossing minimization for visualization problems.

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Section: A Systematic Approachmentioning

confidence: 99%

mentioning

confidence: 99%

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The Graph Crossing Number and its Variants: A Survey

Schaefer

2013

Electron. J. Combin.

100

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“…In [2] the authors prove that n ≤ χ (K n ) ≤ cn 3/2 , for some constant c > 0 and any complete geometric graph K n . See [1] for similar results. Now, we can state an upper bound for the chromatic index of every decomposition of the complete geometric graph.…”

Section: Points In General Positionmentioning

confidence: 53%

Coloring decompositions of complete geometric graphs

2019

Self Cite

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A decomposition of a non-empty simple graph G is a pair [G, P ], such that P is a set of non-empty induced subgraphs of G, and every edge of G belongs to exactly one subgraph in P . The chromatic index χ ([G, P ]) of a decomposition [G, P ] is the smallest number k for which there exists a k-coloring of the elements of P in such a way that: for every element of P all of its edges have the same color, and if two members of P share at least one vertex, then they have different colors. A long standing conjecture of Erdős-Faber-Lovász states that every decomposition [Kn, P ] of the complete graph Kn satisfies χ ([Kn, P ]) ≤ n. In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922.general position. We also consider the particular case when the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.

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“…Aichholzer et. al in [1] have extended the idea of computation of these indices to complete geometric graphs.…”

Section: Introductionmentioning

confidence: 99%

On vertex coloring of graphs

Yegnanarayanan¹,

Logeshwary²

2015

ijma

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The concept of vertex coloring pose a number of challenging open problems in graph theory. Among several interesting parameters, the coloring parameter, namely the pseudoachromatic number of a graph stands a class apart. Although not studied very widely like other parameters in the graph coloring literature, it has started gaining prominence in recent years. The pseudoachromatic number of a simple graph G, denoted ψ(G), is the maximum number of colors used in a vertex coloring of G, where the adjacent vertices may or may not receive the same color but any two distinct pair of colors are represented by at least one edge in it. In this paper we have computed this parameter for a number of classes of graphs.

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Geometric Achromatic and Pseudoachromatic Indices

Cited by 8 publications

References 15 publications

The Graph Crossing Number and its Variants: A Survey

The Graph Crossing Number and its Variants: A Survey

Coloring decompositions of complete geometric graphs

On vertex coloring of graphs

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