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How to decrease the diameter of triangle-free graphs

1998
*
Combinatorica
*

*To*

*the*memory

*of*Paul Erdős Assume that G is a

*triangle*-

*free*

*graph*. Let h d (G) be

*the*minimum number

*of*edges one has

*to*add

*to*G

*to*get a

*graph*

*of*

*diameter*at most d which is still

*triangle*-

*free*. ... It is also shown that

*the*maximum value

*of*h 2 (G) over (

*triangle*-

*free*)

*graphs*

*of*order n is n/2 − 1 n/2 − 1 .

*The*behavior

*of*h 3 (G) is different, its maximum value is n − 1. ... Since an M T F

*graph*is

*of*

*diameter*2, one can generalize

*the*problem above by defining h d (G) as

*the*minimum number

*of*edges needed

*to*extend

*the*

*triangle*-

*free*

*graph*G into a

*triangle*-

*free*

*graph*

*of*

*diameter*...

##
###
Page 6817 of Mathematical Reviews Vol. , Issue 2000j
[page]

2000
*
Mathematical Reviews
*

Mirko Horridk (SK-SFRK-GA; Koiice)
2000j:05061 05C35 0sCi2 Erdés, Paul [Erdés, Paul']; Gyarfas, Andras (H-AOS-C; Budapest); Ruszinké, Miklos (H-AOS-C; Budapest)

*How**to**decrease**the**diameter**of**triangle*-*free*... Let G be a*triangle*-*free**graph**of*order n and set h(G) equal*to**the*minimum number*of*edges that must be added*to*G*to*obtain a maximal*triangle*-*free**graph*. ...##
###
A History of Flips in Combinatorial Triangulations
[chapter]

2012
*
Lecture Notes in Computer Science
*

We provide a comprehensive survey, including full proofs,

doi:10.1007/978-3-642-34191-5_3
fatcat:ntghfi5jqjbqpdr3j7aquyf2ty
*of**the*various attempts*to*answer it. ... Given two combinatorial triangulations,*how*many edge flips are necessary and sufficient*to*convert one into*the*other? This question has occupied researchers for over 75 years. ... However, we focus mainly on attempts*to*determine*the**diameter**of**the*flip*graph*. ...##
###
The dual diameter of triangulations

2018
*
Computational geometry
*

Furthermore, we investigate

doi:10.1016/j.comgeo.2017.06.008
fatcat:vfckb57yjfag5ij3pi5rnafnai
*the*relationship between*the*dual*diameter*and*the*number*of*\emph{ears} (*triangles*with exactly two edges incident*to**the*boundary*of*$\Poly$) in a triangulation. ...*The*\emph{dual*graph*} $\triang^*$*of*a triangulation~$\triang$*of*~$\Poly$ is*the**graph*whose vertices correspond*to**the*bounded faces*of*$\triang$ and whose edges connect those faces*of*~$\triang$ that ... state budget*of**the*Czech Republic. ...##
###
Notes on the betweenness centrality of a graph

2012
*
Mathematica Slovaca
*

We study upper bounds for this invariant and its relations

doi:10.2478/s12175-011-0065-7
fatcat:dlso35wypvcmbjqynn4f5364vy
*to**the**diameter*and average distance*of*a*graph*. ... AbstractThe betweenness centrality*of*a vertex*of*a*graph*is*the*portion*of**the*shortest paths between all pairs*of*vertices passing through a given vertex. ... Nevertheless, if we restrict our study*to**triangle*-*free**graphs*, we obtain*the*complete characterization: Lemma 2. Let G be a*triangle*-*free**graph*. ...##
###
Page 3173 of Mathematical Reviews Vol. , Issue 90F
[page]

1990
*
Mathematical Reviews
*

An undirected

*graph**of**diameter*D is said*to*be D-critical if*the*addition*of*any edge*decreases*its*diameter*. ... These problems are also solved when*the**graph*is assumed*triangle*-*free*and when*the**graph*is assumed C4-*free*.*The*result in*the*final case is surprisingly smaller than in*the*other cases. ...##
###
Algorithms and Almost Tight Results for 3-Colorability of Small Diameter Graphs
[chapter]

2013
*
Lecture Notes in Computer Science
*

on

doi:10.1007/978-3-642-35843-2_29
fatcat:3segyayej5azzpyhuu2eqfpjoa
*triangle*-*free**graphs*with*diameter*3 and minimum degree δ = Θ(n ε ). ... For*graphs*with*diameter*at most 3, we establish*the*complexity*of*3-coloring by proving for every ε ∈ [0, 1) that 3-coloring is NP-complete on*triangle*-*free**graphs**of**diameter*3 and radius 2 with n vertices ...*The*complexity status*of*3-coloring is open also for*triangle*-*free**graphs**of**diameter*2 and*of**diameter*3. ...##
###
Algorithms and Almost Tight Results for $$3$$ 3 -Colorability of Small Diameter Graphs

2014
*
Algorithmica
*

on

doi:10.1007/s00453-014-9949-6
fatcat:rgunvwn2drgxxbv7nhdyf3c6p4
*triangle*-*free**graphs*with*diameter*3 and minimum degree δ = Θ(n ε ). ... For*graphs*with*diameter*at most 3, we establish*the*complexity*of*3-coloring by proving for every ε ∈ [0, 1) that 3-coloring is NP-complete on*triangle*-*free**graphs**of**diameter*3 and radius 2 with n vertices ...*The*complexity status*of*3-coloring is open also for*triangle*-*free**graphs**of**diameter*2 and*of**diameter*3. ...##
###
Page 5474 of Mathematical Reviews Vol. , Issue 2002H
[page]

2002
*
Mathematical Reviews
*

It is shown

*how*a relaxation*of*this problem leads*to*a linear pro- gramming problem and hence*to*a fractional version*of**the*metric dimension*of*a*graph*. ... ; Liu, Yan Pei (PRC-NJT; Beijing) On*the*maximum genus*of**graphs*with*diameter*four. ...##
###
Optimal packings of up to six equal circles on a triangular flat torus

2011
*
Journal of Geometry
*

*The*study

*of*equal circle packings on

*the*standard triangular torus is related

*to*this conjecture. ...

*The*unique globally maximally dense packing

*of*n − 1 circles on

*the*standard triangular torus is congruent

*to*C with one circle removed. ... For a fixed n,

*how*many abstract

*graphs*satisfy

*the*properties in Observation 5.1? 2.

*How*many ways can those abstract

*graphs*from Question 1 be embedded on a flat torus? ...

##
###
Approximate Closest Community Search in Networks
[article]

2015
*
arXiv
*
pre-print

However, most existing studies do not address

arXiv:1505.05956v2
fatcat:73ueznc7kzbmbj4zjdmnxbabs4
*the*"*free*rider" issue, that is, nodes far away from query nodes and irrelevant*to*them are included in*the*detected community. ... In this paper, given an undirected*graph*G and a set*of*query nodes Q, we study community search using*the*k-truss based community model. ... Given*the*recent surge*of*interest in probabilisic*graphs*, an exciting question is*how*k-truss generalizes*to*probabilistic*graphs*. ...##
###
Minimal Prime Graphs of Solvable Groups
[article]

2020
*
arXiv
*
pre-print

Minimal prime

arXiv:2011.10403v1
fatcat:b32grgljnjfl3p3afpxkqev6u4
*graphs*turn out*to*be*of*particular interest, and in this paper we pursue this further by exploring, among other things,*diameters*, Hamiltonian cycles and*the*property*of*being self-complementary ... In finite group theory studying*the*prime*graph**of*a group has been an important topic for*the*past almost half century. ...*The*authors graciously acknowledge*the*financial support*of*NSF, as well as*the*hospitality*of*Texas State University. In particular, Dr. ...##
###
Fast Recoloring of Sparse Graphs
[article]

2014
*
arXiv
*
pre-print

, at each step,

arXiv:1411.6997v1
fatcat:hzs4gu2y7rbaxdp4ozgppf6rza
*the*current coloring is proper. ... We also show that any (2d+2)-coloring*of*a d-degenerate*graph*can be transformed into any other one using a linear number*of*recolorings. ... Acknowledgement*The*authors want*to*thank Marthe Bonamy for fruitful discussions and for pointed out a weaker version*of*Theorem 1. ...##
###
Algorithms and Almost Tight Results for 3-Colorability of Small Diameter Graphs
[article]

2012
*
arXiv
*
pre-print

For

arXiv:1202.4665v3
fatcat:t22xn2fcfrdademfob7fppevnu
*graphs*with*diameter*3 we establish*the*complexity*of*3-coloring by proving that for every ε∈ [0,1), 3-coloring is NP-complete on*triangle*-*free**graphs**of**diameter*3 and radius 2 with n vertices and ... Moreover, assuming ETH, we provide three different amplifications*of*our hardness results*to*obtain for every ε∈ [0,1) subexponential lower bounds for*the*complexity*of*3-coloring on*triangle*-*free**graphs*...*The*complexity status*of*3-coloring is open also for*triangle*-*free**graphs**of**diameter*2 and*of**diameter*3. ...##
###
The Price of Anarchy of a Network Creation Game with Exponential Payoff
[chapter]

2008
*
Lecture Notes in Computer Science
*

This allows us

doi:10.1007/978-3-540-79309-0_20
fatcat:khib4iagl5eodjls55zasf7xpi
*to*thoroughly reveal*the*structure*of*all stable states. In addition,*the*question for*the*price*of*anarchy can be reduced*to*counting*the*maximum number*of*edges*of*a stable*graph*. ... We show that*the*process has a positive probability*to*cycle. We reduce*the*creation rule with payoff functions*to**graph*theoretic criteria. Moreover, these criteria can be evaluated locally. ... Intuitively, stable*graphs*with*triangles*should have more edges than*triangle**free*stable*graphs*. ...
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