1
$\begingroup$

For each $n\geq1$, consider a special tree with $2n+1$ nodes which are assigned values $a_i$ from the set $\{0,0,1,2,3,\dots,2n-1\}$. Only $0$ can be a repeated assignment.

The edges are only the ones connecting $a_{2i}$ to $a_{2i+2}$ and $a_{2i+1}$ to $a_{2i+2}$. Then, the edge connecting $x$ and $y$ has weight $x+y$.

Here are some figures for an illustrative purpose for the special trees we focus on.

GOAL: Make the edge-weights $a_i+a_j$ mutually distinct modulo $2n$.

Example: if $2n+1=3$ then $a_0=a_2=0, a_1=1$ works, mod $2$.

Example: if $2n+1=5$ then $a_0=a_3=0, a_1=1, a_2=3, a_4=2$, mod $4$.

QUESTION. Is it always possible to real the above goal?

Improve this question
$\endgroup$
7
  • $\begingroup$ ?? If there's an edge connecting $a_i$ and $a_j$ for each $i$ and $j$, then that's not a tree. Can you tell us here what the trees are, instead of making us go offsite to find out? $\endgroup$
    – Gerry Myerson
    Commented Apr 25, 2019 at 5:09
  • 1
    $\begingroup$ @GerryMyerson: I've added a few lines to clarify this. $\endgroup$
    – T. Amdeberhan
    Commented Apr 25, 2019 at 18:42
  • $\begingroup$ So, the $a_i$ are both the values of the nodes, and the names of the nodes? And the node named $a_{2n-1}$ is isolated? $\endgroup$
    – Gerry Myerson
    Commented Apr 25, 2019 at 22:32
  • $\begingroup$ Yes, both the value and the name. I was hoping the pictures in the link shade more light into it, if you can view them. $\endgroup$
    – T. Amdeberhan
    Commented Apr 25, 2019 at 23:47
  • $\begingroup$ So there are two nodes with the name $a_0$? $\endgroup$
    – Gerry Myerson
    Commented Apr 26, 2019 at 3:28

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .