• Corpus ID: 245650242

The extremality of 2-partite Tur\'an graphs with respect to the number of colorings

@inproceedings{Fuentes2021TheEO,
  title={The extremality of 2-partite Tur\'an graphs with respect to the number of colorings},
  author={Melissa M Fuentes},
  year={2021}
}
We consider a problem proposed by Linial and Wilf to determine the structure of graphs which allows the maximum number of q-colorings among graphs with n vertices and m edges. Let Tr(n) denote the Turán graph the complete r-partite graph on n vertices with partition sizes as equal as possible. We prove that for all odd integers q ≥ 5 and sufficiently large n, the Turán graph T2(n) has at least as many q-colorings as any other graph G with the same number of vertices and edges as T2(n), with… 

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References

SHOWING 1-10 OF 26 REFERENCES
Turán Graphs and the Number of Colorings
TLDR
It is shown that if r divides qu, then for all sufficiently large n, the Turan graph T_r(n) has more qu-colorings than any other graph with the same number of vertices and edges.
Maximizing proper colorings on graphs
The maximum number of colorings of graphs of given order and size: A survey
Maximizing the number of q‐colorings
Let PG(q) denote the number of proper q‐colorings of a graph G. This function, called the chromatic polynomial of G, was introduced by Birkhoff in 1912, who sought to attack the famous four‐color
Maximum number of colorings of (2k, k 2 )-graphs
Let ℱ2k, k2 consist of all simple graphs on 2k vertices and k2 edges. For a simple graph G and a positive integer λ, let PG(λ) denote the number of proper vertex colorings of G in at most λ colors,
An Extremal Property of Turán Graphs, II
TLDR
It is shown that for all , Turan's graph has more proper vertex colorings in at most 4 colors than any other graph with the same number of vertices and edges.
Some new bounds for the maximum number of vertex colorings of a (v, e)-graph
TLDR
A new notion of pseudoproper colorings of a graph is given, which allows us to significantly improve the upper bounds for f(v, e, 3) given by Lazebnik and Liu in the case where e > v2/4.
New upper bounds for the greatest number of proper colorings of a (V, E)-graph
TLDR
Some new upper bounds are determined for the greatest number of proper colorings in λ or less colors that a (υ,e)-graph G can have, i.e. f(λ, e,λ) =max{P(G;λ): G∈F}.
Some corollaries of a theorem of Whitney on the chromatic polynomial
A Theoretical Analysis of Backtracking in the Graph Coloring Problem
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