We are given a $n$-partite weighted graph $G$. Each partition has $n$ vertices, some of which may be isolated. Each partition must contain at least one non-isolated vertex. Let us number the vertices in some $i^{th}$ partition as $V_{i1},V_{i2},...,V_{in}$. Now each non-isolated vertex $V_{ij}$ has a set of $n-1$ neighbors (one in each of the remaining $n-1$ partitions) that form a permutation with $j$. Vertex $V_{ij}$ can have other neighbors as well. In fact it can have several such sets of neighbors.
Every non-isolated vertex has a positive weight $w_{ij}$ and every edge $(V_{ij},V_{kl})$ has a positive weight $e_{ijkl}$. For every non-isolated vertex $V_{ij}$, the following must be true $w_{ij}=\sum_{l}e_{ijkl}\forall k(\neq i)=\sum_{k}e_{ijkl}\forall l$l(\neq j)$. As a consequence of this condition, if a vertex has only one neighbor at position$l$(of some partition$k$) among all partitions, then it cannot have any other neighbor in that partition$k$. Also, if a vertex has only one neighbor in some partition$k$(at some position$l$), then it cannot have any neighbor at that position in any other partition. We conjecture that the graph$G$will always contain a$n$-clique. Is it true? 3 fixed an equation We are given a$n$-partite weighted graph$G$. Each partition has$n$vertices, some of which may be isolated. Each partition must contain at least one non-isolated vertex. Let us number the vertices in some$i^{th}$partition as$V_{i1},V_{i2},...,V_{in}$. Now each non-isolated vertex$V_{ij}$has a set of$n-1$neighbors (one in each of the remaining$n-1$partitions) that form a permutation with$j$. Vertex$V_{ij}$can have other neighbors as well. In fact it can have several such sets of neighbors. Every non-isolated vertex has a positive weight$w_{ij}$and every edge$(V_{ij},V_{kl})$has a positive weight$e_{ijkl}$. For every non-isolated vertex$V_{ij}$, the following must be true$w_{ij}=\sum_{l}e_{ijkl}\forall k=\sum_{k}e_{ijkl}\forall k(\neq i)=\sum_{k}e_{ijkl}\forall l$. As a consequence of this condition, if a vertex has only one neighbor at position$l$(of some partition$k$) among all partitions, then it cannot have any other neighbor in that partition$k$. Also, if a vertex has only one neighbor in some partition$k$(at some position$l$), then it cannot have any neighbor at that position in any other partition. We conjecture that the graph$G$will always contain a$n$-clique. Is it true? 2 added text We are given a$n$-partite weighted graph$G$. Each partition has$n$vertices, some of which may be isolated. Each partition must contain at least one non-isolated vertex. Let us number the vertices in some$i^{th}$partition as$V_{i1},V_{i2},...,V_{in}$. Now each non-isolated vertex$V_{ij}$has a set of$n-1$neighbors (one in each of the remaining$n-1$partitions) that form a permutation with$j$. Vertex$V_{ij}$can have other neighbors as well. In fact it can have several such sets of neighbors. Every non-isolated vertex has a positive weight$w_{ij}$and every edge$(V_{ij},V_{kl})$has a positive weight$e_{ijkl}$. For every non-isolated vertex$V_{ij}$, the following must be true$w_{ij}=\sum_{l}e_{ijkl}\forall k=\sum_{k}e_{ijkl}\forall l$. As a consequence of this condition, if a vertex has only one neighbor at position$l$(of some partition$k$) among all partitions, then it cannot have any other neighbor in that partition$k$. Also, if a vertex has only one neighbor in some partition$k$(at some position$l$), then it cannot have any neighbor at that position in any other partition. We conjecture that the graph$G$will always contain a$n\$-clique. Is it true?