scholarly journals On θ-commutators and the corresponding non-commuting graphs

2017 ◽  
Vol 15 (1) ◽  
pp. 1530-1538
Author(s):  
S. Shalchi ◽  
A. Erfanian ◽  
M. Farrokhi DG
Keyword(s):  
Chromatic Number ◽  
Independent Sets ◽  
Graphs Of Groups

Abstract The θ-commutators of elements of a group with respect to an automorphism are introduced and their properties are investigated. Also, corresponding to θ-commutators, we define the θ-non-commuting graphs of groups and study their correlations with other notions. Furthermore, we study independent sets in θ-non-commuting graphs, which enable us to evaluate the chromatic number of such graphs.

The Strong Perfect Graph Conjecture for Planar Graphs

1973 ◽  
Vol 25 (1) ◽  
pp. 103-114 ◽  
Author(s):  
Alan Tucker
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Independent Set ◽  
Independent Sets ◽  
Clique Number ◽  
Perfect Graph ◽  
Stability Number ◽  
Partition Number ◽  
Minimum Number ◽  
The Stability

A graph G is called γ-perfect if ƛ (H) = γ(H) for every vertex-generated subgraph H of G. Here, ƛ(H) is the clique number of H (the size of the largest clique of H) and γ(H) is the chromatic number of H (the minimum number of independent sets of vertices that cover all vertices of H). A graph G is called α-perfect if α(H) = θ(H) for every vertex-generated subgraph H of G, where α (H) is the stability number of H (the size of the largest independent set of H) and θ(H) is the partition number of H (the minimum number of cliques that cover all vertices of H).

scholarly journals Symmetric Graphs with Respect to Graph Entropy

10.37236/5642 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Seyed Saeed Changiz Rezaei ◽  
Ehsan Chiniforooshan
Keyword(s):  
Probability Distribution ◽  
Chromatic Number ◽  
Probability Distributions ◽  
Independence Number ◽  
Maximum Size ◽  
Independent Sets ◽  
Fractional Chromatic Number ◽  
Vertex Set ◽  
Definition Of ◽  
The Relationship

Let $F_G(P)$ be a functional defined on the set of all the probability distributions on the vertex set of a graph $G$. We say that $G$ is symmetric with respect to $F_G(P)$ if the uniform distribution on $V(G)$ maximizes $F_G(P)$. Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic number, we characterize all graphs which are symmetric with respect to graph entropy. We show that a graph is symmetric with respect to graph entropy if and only if its vertex set can be uniformly covered by its maximum size independent sets. This is also equivalent to saying that the fractional chromatic number of $G$, $\chi_f(G)$, is equal to $\frac{n}{\alpha(G)}$, where $n = |V(G)|$ and $\alpha(G)$ is the independence number of $G$. Furthermore, given any strictly positive probability distribution $P$ on the vertex set of a graph $G$, we show that $P$ is a maximizer of the entropy of graph $G$ if and only if its vertex set can be uniformly covered by its maximum weighted independent sets. We also show that the problem of deciding if a graph is symmetric with respect to graph entropy, where the weight of the vertices is given by probability distribution $P$, is co-NP-hard.

scholarly journals Chromatic Number in Time O(2.4023^n) Using Maximal Independent Sets

2002 ◽  
Vol 9 (45) ◽  
Author(s):  
Jesper Makholm Byskov
Keyword(s):  
Chromatic Number ◽  
Independent Sets ◽  
Running Time ◽  
Maximal Independent Sets

In this paper we improve an algorithm by Eppstein (2001) for finding the chromatic number of a graph. We modify the algorithm slightly, and by using a bound on the number of maximal independent sets of size  k from our recent paper (2003), we prove that the running time is O(2.4023^n). Eppstein's algorithm runs in time O(2.4150^n). The space usage for both algorithms is O(2^n).

scholarly journals Estimating the fractional chromatic number of a graph

2021 ◽  
Vol 13 (1) ◽  
pp. 122-133
Author(s):  
Sándor Szabó
Keyword(s):  
Chromatic Number ◽  
Numerical Experiments ◽  
Independent Sets ◽  
Linear Program ◽  
Fractional Chromatic Number ◽  
Upper Estimate ◽  
The Given

Abstract The fractional chromatic number of a graph is defined as the optimum of a rather unwieldy linear program. (Setting up the program requires generating all independent sets of the given graph.) Using combinatorial arguments we construct a more manageable linear program whose optimum value provides an upper estimate for the fractional chromatic number. In order to assess the feasibility of the proposal and in order to check the accuracy of the estimates we carry out numerical experiments.

scholarly journals Maximum Independent Sets Partition of (n,k)-Star Graphs

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Fu-Tao Hu
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Computer Modelling ◽  
Independent Sets ◽  
Star Graph ◽  
Star Graphs ◽  
Independent Number ◽  
Two Parameters

The (n,k)-star graph is a very important computer modelling. The independent number and chromatic number of a graph are two important parameters in graph theory. However, we have not known the values of these two parameters of the (n,k)-star graph since it was proposed. In this paper, we show a maximum independent sets partition of (n,k)-star graph. From that, we can immediately deduce the exact value of the independent number and chromatic number of (n,k)-star graph.

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