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Recognizable colorings of cycles and trees

Michael J. Dorfling, Samantha Dorfling (2012)

Discussiones Mathematicae Graph Theory

For a graph G and a vertex-coloring c:V(G) → 1,2, ...,k, the color code of a vertex v is the (k+1)-tuple (a₀,a₁, ...,aₖ), where a₀ = c(v), and for 1 ≤ i ≤ k, a i is the number of neighbors of v colored i. A recognizable coloring is a coloring such that distinct vertices have distinct color codes. The recognition number of a graph is the minimum k for which G has a recognizable k-coloring. In this paper we prove three conjectures of Chartrand et al. in [8] regarding the recognition number of cycles...

Recognizable colorings of graphs

Gary Chartrand, Linda Lesniak, Donald W. VanderJagt, Ping Zhang (2008)

Discussiones Mathematicae Graph Theory

Let G be a connected graph and let c:V(G) → 1,2,...,k be a coloring of the vertices of G for some positive integer k (where adjacent vertices may be colored the same). The color code of a vertex v of G (with respect to c) is the ordered (k+1)-tuple code(v) = (a₀,a₁,...,aₖ) where a₀ is the color assigned to v and for 1 ≤ i ≤ k, a i is the number of vertices adjacent to v that are colored i. The coloring c is called recognizable if distinct vertices have distinct color codes and the recognition number...

Rectangular table negotiation problem revisited

Dalibor Froncek, Michael Kubesa (2011)

Open Mathematics

We solve the last missing case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Suppose we have two negotiating delegations with n=mk members each and we have a seating arrangement such that every day the negotiators sit at m tables with k people of the same delegation at one side of each table. Every person can effectively communicate just with three nearest persons across the table. Our goal is to guarantee that over the course of several days,...

Repetition thresholds for subdivided graphs and trees

Pascal Ochem, Elise Vaslet (2012)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

The repetition threshold introduced by Dejean and Brandenburg is the smallest real number α such that there exists an infinite word over a k-letter alphabet that avoids β-powers for all β > α. We extend this notion to colored graphs and obtain the value of the repetition thresholds of trees and “large enough” subdivisions of graphs for every alphabet size.

Repetition thresholds for subdivided graphs and trees

Pascal Ochem, Elise Vaslet (2012)

RAIRO - Theoretical Informatics and Applications

The repetition threshold introduced by Dejean and Brandenburg is the smallest real number α such that there exists an infinite word over a k-letter alphabet that avoids β-powers for all β > α. We extend this notion to colored graphs and obtain the value of the repetition thresholds of trees and “large enough” subdivisions of graphs for every alphabet size.

Set colorings in perfect graphs

Ralucca Gera, Futaba Okamoto, Craig Rasmussen, Ping Zhang (2011)

Mathematica Bohemica

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 ∈ V ( 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 χ s ( G ) . We show that the decision variant of determining χ s ( G ) is NP-complete in the general case, and show that χ s ( G ) can be...

Sharp bounds for the number of matchings in generalized-theta-graphs

Ardeshir Dolati, Somayyeh Golalizadeh (2012)

Discussiones Mathematicae Graph Theory

A generalized-theta-graph is a graph consisting of a pair of end vertices joined by k (k ≥ 3) internally disjoint paths. We denote the family of all the n-vertex generalized-theta-graphs with k paths between end vertices by Θⁿₖ. In this paper, we determine the sharp lower bound and the sharp upper bound for the total number of matchings of generalized-theta-graphs in Θⁿₖ. In addition, we characterize the graphs in this class of graphs with respect to the mentioned bounds.

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