the brown wash-rag of a back
I know what "wash-rag" is.
But I don't understand the meaning of “the brown wash-rag of a back” in the excerpt below. Is it a common expression?
[I’d like to get one of those police dogs; I don’t suppose you
got that kind?”
The man peered doubtfully into the basket, plunged in his hand
and drew one up, wriggling, by the back of the neck.
“That’s no police dog,” said Tom.
“No, it’s not exactly a polICE dog,” said the man with
disappointment in his voice. “It’s more of an Airedale.” He passed
his hand over the brown wash-rag of a back.]