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### How to decrease the diameter of triangle-free graphs

Paul Erdős, András Gyárfás, Miklós Ruszinkó
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]

Prosenjit Bose, Sander Verdonschot
2012 Lecture Notes in Computer Science
We provide a comprehensive survey, including full proofs, 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

Matias Korman, Stefan Langerman, Wolfgang Mulzer, Alexander Pilz, Maria Saumell, Birgit Vogtenhuber
2018 Computational geometry
Furthermore, we investigate 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

S. Gago, J. Hurajová, T. Madaras
2012 Mathematica Slovaca
We study upper bounds for this invariant and its relations 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]

George B. Mertzios, Paul G. Spirakis
2013 Lecture Notes in Computer Science
on 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

George B. Mertzios, Paul G. Spirakis
2014 Algorithmica
on 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

William Dickinson, Daniel Guillot, Anna Keaton, Sandi Xhumari
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]

Xin Huang, Laks V.S. Lakshmanan, Jeffrey Xu Yu, Hong Cheng
2015 arXiv   pre-print
However, most existing studies do not address 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]

Chris Florez, Jonathan Higgins, Kyle Huang, Thomas Michael Keller, Dawei Shen
2020 arXiv   pre-print
Minimal prime 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]

Nicolas Bousquet, Guillem Perarnau
2014 arXiv   pre-print
, at each step, 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]

George B. Mertzios, Paul G. Spirakis
2012 arXiv   pre-print
For 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.  ...