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Combinatorial problem on subgraphs of the Johnson graph
Posted:
Sep 20, 2012 6:00 AM


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...



