scholarly journals THE CHARACTER GRAPH OF A FINITE GROUP IS PERFECT

2020 ◽  
pp. 1-5
Author(s):  
MAHDI EBRAHIMI
Keyword(s):  
Finite Group ◽  
Chromatic Number ◽  
Clique Number ◽  
Perfect Graph ◽  
Induced Subgraph ◽  
Delta G ◽  
A Finite Group ◽  
Complex Characters

Abstract For a finite group G, let $\Delta (G)$ denote the character graph built on the set of degrees of the irreducible complex characters of G. A perfect graph is a graph $\Gamma $ in which the chromatic number of every induced subgraph $\Delta $ of $\Gamma $ equals the clique number of $\Delta $ . We show that the character graph $\Delta (G)$ of a finite group G is always a perfect graph. We also prove that the chromatic number of the complement of $\Delta (G)$ is at most three.

scholarly journals A Strengthening of Brooks' Theorem for Line Graphs

10.37236/632 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Clique Number ◽  
Line Graphs ◽  
Delta G

We prove that if $G$ is the line graph of a multigraph, then the chromatic number $\chi(G)$ of $G$ is at most $\max\left\{\omega(G), \frac{7\Delta(G) + 10}{8}\right\}$ where $\omega(G)$ and $\Delta(G)$ are the clique number and the maximum degree of $G$, respectively. Thus Brooks' Theorem holds for line graphs of multigraphs in much stronger form. Using similar methods we then prove that if $G$ is the line graph of a multigraph with $\chi(G) \geq \Delta(G) \geq 9$, then $G$ contains a clique on $\Delta(G)$ vertices. Thus the Borodin-Kostochka Conjecture holds for line graphs of multigraphs.

scholarly journals Induced Subgraphs of Graphs with Large Chromatic Number IX: Rainbow Paths

10.37236/6768 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Alex Scott ◽  
Paul Seymour
Keyword(s):  
Recent Result ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Large Chromatic Number ◽  
Vertex Colouring

We prove that for all integers $\kappa, s\ge 0$ there exists $c$ with the following property. Let $G$ be a graph with clique number at most $\kappa$ and chromatic number more than $c$. Then for every vertex-colouring (not necessarily optimal) of $G$, some induced subgraph of $G$ is an $s$-vertex path, and all its vertices have different colours. This extends a recent result of Gyárfás and Sárközy (2016) who proved the same for graphs $G$ with $\kappa=2$ and girth at least five.

scholarly journals A new graph over semi-direct products of groups

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 611-619
Author(s):  
Sercan Topkaya ◽  
Sinan Cevik
Keyword(s):  
Chromatic Number ◽  
Final Section ◽  
Degree Sequence ◽  
Domination Number ◽  
Clique Number ◽  
Perfect Graph ◽  
Direct Products ◽  
Products Of Groups ◽  
Minimum Degrees ◽  
Product Of Groups

In this paper, by establishing a new graph ?(G) over the semi-direct product of groups, we will first state and prove some graph-theoretical properties, namely, diameter, maximum and minimum degrees, girth, degree sequence, domination number, chromatic number, clique number of ?(G). In the final section we will show that ?(G) is actually a perfect graph.

RATIONAL NEARLY SIMPLE GROUPS

2020 ◽  
pp. 1-11
Author(s):  
FARIDEH SHAFIEI ◽  
MOHAMMAD REZA DARAFSHEH ◽  
FARROKH SHIRJIAN
Keyword(s):  
Finite Group ◽  
Simple Groups ◽  
Rational Group ◽  
A Finite Group ◽  
Complex Characters

Abstract A finite group whose irreducible complex characters are rational-valued is called a rational group. The aim of this paper is to determine the rational almost simple and rational quasi-simple groups.

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 Tensor Product Of Zero-divisor Graphs With Finite Free Semilattices

2017 ◽  
Vol 9 (1) ◽  
pp. 13
Author(s):  
Kemal Toker
Keyword(s):  
Tensor Product ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Independence Number ◽  
Domination Number ◽  
Clique Number ◽  
Perfect Graph ◽  
Chromatic Index

$\Gamma (SL_{X})$ is defined and has been investigated in (Toker, 2016). In this paper our main aim is to extend this study over  $\Gamma (SL_{X})$ to the tensor product. The diameter, radius, girth, domination number, independence number, clique number, chromatic number and chromatic index of $\Gamma (SL_{X_{1}})\otimes \Gamma (SL_{X_{2}})$ has been established. Moreover, we have determined when $\Gamma (SL_{X_{1}})\otimes \Gamma (SL_{X_{2}})$ is a perfect graph.

scholarly journals The independence graph of a finite group

2020 ◽  
Vol 193 (4) ◽  
pp. 845-856
Author(s):  
Andrea Lucchini
Keyword(s):  
Finite Group ◽  
Planar Graph ◽  
Generating Set ◽  
Delta G ◽  
A Finite Group ◽  
Minimal Generating Set

Abstract Given a finite group G, we denote by $$\Delta (G)$$ Δ ( G ) the graph whose vertices are the elements G and where two vertices x and y are adjacent if there exists a minimal generating set of G containing x and y. We prove that $$\Delta (G)$$ Δ ( G ) is connected and classify the groups G for which $$\Delta (G)$$ Δ ( G ) is a planar graph.

scholarly journals On Graphs Related to Comaximal Ideals of a Commutative Ring

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Tongsuo Wu ◽  
Meng Ye ◽  
Dancheng Lu ◽  
Houyi Yu
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraph ◽  
The Core ◽  
Maximal Ideals ◽  
Graph Properties ◽  
Vertex Set

We study the co maximal graph Ω(R), the induced subgraph Γ(R) of Ω(R) whose vertex set is R∖(U(R)∪J(R)), and a retract Γr(R) of Γ(R), where R is a commutative ring. For a graph Γ(R) which contains a cycle, we show that the core of Γ(R) is a union of triangles and rectangles, while a vertex in Γ(R) is either an end vertex or a vertex in the core. For a nonlocal ring R, we prove that both the chromatic number and clique number of Γ(R) are identical with the number of maximal ideals of R. A graph Γr(R) is also introduced on the vertex set {Rx∣x∈R∖(U(R)∪J(R))}, and graph properties of Γr(R) are studied.

scholarly journals On the clique number of the generating graph of a finite group

2009 ◽  
Vol 137 (10) ◽  
pp. 3207-3207 ◽  
Author(s):  
Andrea Lucchini ◽  
Attila Maróti
Keyword(s):  
Finite Group ◽  
Clique Number ◽  
Generating Graph ◽  
A Finite Group

scholarly journals On the chromatic number of the power graph of a finite group

2015 ◽  
Vol 26 (4) ◽  
pp. 626-633 ◽  
Author(s):  
Xuanlong Ma ◽  
Min Feng
Keyword(s):  
Finite Group ◽  
Chromatic Number ◽  
Power Graph ◽  
A Finite Group
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