Date: Sep 20, 2012 6:00 AM
Author: Rodolfo Conde
Subject: Combinatorial problem on subgraphs of the Johnson graph
Let $1\leqslant m \leqslant n$, $N=\{ 1,\ldots,n\}$. The Johnson graph

$J_{n,m}$ has as vertex set all subsets of $N$ of cardinality $m$, two

vertices $b_1,b_2$ are adjacent if and only if $\lvert b_1 \cap b_2 \rvert =

m - 1$. Let $V_{n,m}=V(J_{n,m})$. If $U\subset V_{n,m}$, define the set

$\zeta(U)$

as

$\zeta(U)= \{ c\cup d \mid c,d \in U \wedge \lvert c \cap d \rvert = m - 1

\}$.

Notice that each $f\in \zeta(U)$ has size $m+1$, because, if $f= c\cup d$

for $c,d \in U$, then

$\lvert c \rvert = \lvert d \rvert = m$ and it is known that $\lvert c \cap

d \rvert = m - 1

\Leftrightarrow \lvert c \cup d \rvert = m + 1$. Thus $\zeta(U)\subseteq

V_{n,m+1}$.

The problem: Suppose that $U\subset 2^N$ is a collection of subsets of $N$,

all of them of size m. Assume also that $\lvert U \rvert \leqslant n - m$

and that the subgraph $G\left[ U \right]$ of $J_{n,m}$ induced by $U$ is

disconnected.

Then prove that $\bigcup_{b\in \zeta(U)}b \neq N \vee G\left[ \zeta(U)

\right]$ is a disconnected subgraph of $J_{n,m+1}$.

The hard part of the proof, in which i have been unsuccessful, is when we

suppose that $\bigcup_{b\in \zeta(U)}b = N$, we have to show that $G\left[

\zeta(U) \right]$ is disconnected.

Can anyone shed some light on this matter ? Any hints or references ?

Thanks a lot for all your comments !!!

Greetings...