Bounds on vertex colorings with restrictions on the union of color classes

Aravind, N. R.; Subramanian, C. R.

doi:10.1002/jgt.20501

Cited by 10 publications

(30 citation statements)

References 21 publications

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“…Here, m denotes the minimum number of edges in any member of F . This bound is also nearly tight in view of the lower bounds in derived in [22] for the case j = 2. In this work, a connection between (j, F )-chromatic numbers and oriented chromatic numbers was also established.…”

Section: Constrained Coloringsmentioning

confidence: 65%

“…Since these events are independent for different pairs (involving disjoint edges), we deduce that 1/2 . This proof is a simplification of the proof of a more general theorem which can be found in [22], the latter proof using arguments similar to those employed in [7].…”

Section: Establishing Tightnessmentioning

confidence: 94%

“…In a subsequent work [23], the authors improved the bound of [22] (for the special case of j = 2) to O d m m−1 . Here, m denotes the minimum number of edges in any member of F .…”

Section: Constrained Coloringsmentioning

confidence: 96%

“…Recently, Aravind and Subramanian [22] have generalized the notions of acyclic chromatic number, frugal chromatic number, etc. to a generic notion of (j, F )-colorings and and obtained upper bounds (in terms of Δ) on the associated chromatic numbers.…”

Section: Constrained Coloringsmentioning

confidence: 99%

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Probabilistic Arguments in Graph Coloring (Invited Talk)

Subramanian¹

2015

Algorithms and Discrete Applied Mathematics

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Abstract. Probabilistic arguments has come to be a powerful way to obtain bounds on various chromatic numbers. It plays an important role not only in obtaining upper bounds but also in establishing the tightness of these upper bounds. It often calls for the application of various (often simple) ideas, tools and techniques (from probability theory) like moments, concentration inequalities, known estimates on tail probabilities and various other probability estimates. A number of examples illustrate how this approach can be a very useful tool in obtaining chromatic bounds. For many of these bounds, no other approach for obtaining them is known so far. In this talk, we illustrate this approach with some specific applications to graph coloring. On the other hand, several specific applications of this approach have also motivated and led to the development of powerful tools for handling discrete probability spaces. An important tool (for establishing the tightness results) is the notion of random graphs. We do not necessarily present the best results (obtained using this approach) since the main purpose is to provide an introduction to the power and simplicity of the approach. Many of the examples and the results are already known and published in the literature and have also been improved further.

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Section: Constrained Coloringsmentioning

confidence: 65%

Section: Establishing Tightnessmentioning

confidence: 94%

“…In a subsequent work [23], the authors improved the bound of [22] (for the special case of j = 2) to O d m m−1 . Here, m denotes the minimum number of edges in any member of F .…”

Section: Constrained Coloringsmentioning

confidence: 96%

Section: Constrained Coloringsmentioning

confidence: 99%

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Probabilistic Arguments in Graph Coloring (Invited Talk)

Subramanian¹

2015

Algorithms and Discrete Applied Mathematics

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“…An acyclic coloring of a graph is a proper vertex coloring in which the subgraph induced by the union of any two color classes is a forest. The following definition from [5] generalizes the notion of acyclic coloring, and is related to intersection dimension, as we shall prove later.…”

Section: Definitions and Factsmentioning

confidence: 97%

Intersection dimension and graph invariants

Aravind¹,

Subramanian²

2021

Discuss. Math. Graph Theory

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We show that the intersection dimension of graphs with respect to several hereditary properties can be bounded as a function of the maximum degree. As an interesting special case, we show that the circular dimension of a graph with maximum degree ∆ is at most O ∆ log ∆ log log ∆. It is also shown that permutation dimension of any graph is at most ∆(log ∆) 1+o(1). We also obtain bounds on intersection dimension in terms of treewidth.

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Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings

Bradshaw

2022

Journal of Graph Theory

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We show that for any fixed integer ≥ m 1, a graph of maximum defiggree Δ has a coloring with ∕ O (Δ ) m m ( +1) colors in which every connected bicolored subgraph contains at most m edges. This result unifies previously known upper bounds on the number of colors sufficient for certain types of graph colorings, including star colorings, for which ∕ O (Δ ) 3 2 colors suffice, and acyclic colorings, for which ∕ O (Δ ) 4 3 colors suffice. Our proof uses a probabilistic

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Bounds on vertex colorings with restrictions on the union of color classes

Abstract: A proper coloring of a graph is a labeled partition of its vertices into parts which are independent sets.

Cited by 10 publications

References 21 publications

Probabilistic Arguments in Graph Coloring (Invited Talk)

Probabilistic Arguments in Graph Coloring (Invited Talk)

Intersection dimension and graph invariants

Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings

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