Unit-distance graphs, graphs on the integer lattice and a Ramsey type result

1995 ◽  
Vol 49 (1) ◽  
pp. 319-319
Author(s):  
Kiran B. Chilakamarri ◽  
Carolyn R. Mahoney
Keyword(s):  
Integer Lattice ◽  
Type Result ◽  
Unit Distance ◽  
Distance Graphs ◽  
Ramsey Type

Unit-distance graphs, graphs on the integer lattice and a Ramsey type result

1996 ◽  
Vol 51 (1-2) ◽  
pp. 48-67
Author(s):  
Kiran B. Chilakamarri ◽  
Carolyn R. Mahoney
Keyword(s):  
Integer Lattice ◽  
Type Result ◽  
Unit Distance ◽  
Distance Graphs ◽  
Ramsey Type

scholarly journals ?-Unit Distance Graphs

2004 ◽  
Vol 33 (1) ◽  
pp. 117-123 ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Unit Distance ◽  
Distance Graphs

On the connectivity of unit distance graphs

1996 ◽  
Vol 12 (3) ◽  
pp. 295-303 ◽  
Author(s):  
Michael Reid
Keyword(s):  
Unit Distance ◽  
Distance Graphs

A RAMSEY TYPE RESULT FOR LATIN SQUARES

2019 ◽  
Vol 101 (3) ◽  
pp. 362-366
Author(s):  
YEVHEN ZELENYUK ◽  
YULIYA ZELENYUK
Keyword(s):  
Cartesian Product ◽  
Latin Square ◽  
Latin Squares ◽  
Multiplication Table ◽  
The Other ◽  
Type Result ◽  
Other Hand ◽  
Ramsey Type

We show that for all $m,k,r\in \mathbb{N}$, there is an $n\in \mathbb{N}$ such that whenever $L$ is a Latin square of order $m$ and the Cartesian product $L^{n}$ of $n$ copies of $L$ is $r$-coloured, there is a monochrome Latin subsquare of $L^{n}$, isotopic to $L^{k}$. In particular, for every prime $p$ and for all $k,r\in \mathbb{N}$, there is an $n\in \mathbb{N}$ such that whenever the multiplication table $L({\mathbb{Z}_{p}}^{n})$ of the group ${\mathbb{Z}_{p}}^{n}$ is $r$-coloured, there is a monochrome Latin subsquare of order $p^{k}$. On the other hand, we show that for every group $G$ of order $\leq 15$, there is a 2-colouring of $L(G)$ without a nontrivial monochrome Latin subsquare.

On the span and extent of unit-distance graphs in the plane

2016 ◽  
Vol 10 ◽  
pp. 1611-1618
Author(s):  
Severino V. Gervacio ◽  
Hiroshi Maehara ◽  
Joselito A. Uy
Keyword(s):  
Unit Distance ◽  
Distance Graphs

A Ramsey-type result for the hypercube

2006 ◽  
Vol 53 (3) ◽  
pp. 196-208 ◽  
Author(s):  
Noga Alon ◽  
Radoš Radoičić ◽  
Benny Sudakov ◽  
Jan Vondrák
Keyword(s):  
Type Result ◽  
Ramsey Type

scholarly journals A Characterization in Zn of Finite Unit-Distance Graphs in Rn

1993 ◽  
Vol 59 (1) ◽  
pp. 156-160 ◽  
Author(s):  
K.B. Chilakamarri
Keyword(s):  
Unit Distance ◽  
Distance Graphs
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