New families of graceful disconnected graphs. (Q2715990)

A graph \(G\) is called pseudograceful if there exists an injective function NEWLINE\[NEWLINEf: V(G) \to \{0,1,\dots ,| E(G)| -1,| E(G)| +1\}NEWLINE\]NEWLINE such that the induced function \(f^*\colon E(G) \to \{1,2,\dots ,| E(G)| \}\) defined by \(f^*(xy)=| f(x)-f(y)| \) for all \(xy\in E(G)\) is an injection. This concept is in general incomparable with the notion of a graceful graph. The main theorem states that if \(G\) is pseudograceful, then \(G\cup K_{m,n}\) is also pseudograceful if \(m,n\geq 2\) and \((m,n)\not =(2,2)\), and is graceful if \(m,n\geq 2\). Some new families of graceful disconnected graphs are determined.