Totally Magic Injections of Graphs

For a simple graph \( G \), consider an injection \( \mu: V \cup E \rightarrow \mathbb{N} \). If for every vertex \( x \in V \) we have \( \mu(x) + \sum_{y \sim x} \mu(xy) = h \), and for every edge \( xy \in E(G) \) we have \( \mu(x) + \mu(xy) + \mu(y) = k \), for some constants \( h \) and \( k \), then \( \mu \) is a totally magic injection (TMI) of \( G \). Also, \( m_t(G) \) is the smallest number in \( \mathbb{N} \) such that there is a TMI \( \mu: V \cup E \rightarrow \{1, 2, \ldots, m_t(G)\} \). Here we study TMIs and the number \( m_t(G) \) for certain \( G \). One theorem, the Star Theorem, is useful for eliminating many classes of well-known graphs that could have a TMI. For most \( n \) and \( n_j \), the following graphs do not have a TMI: every non-star tree, \( P_n \), \( C_n \), \( W_n \), \( K_n \), and \( K_{n_1, n_2, \ldots, n_p} \). We determine \( m_t(F) \) for every forest \( F \) that has a TMI, and \( m_t(G) \) for every graph \( G \) with \( \leq 6 \) vertices that has a TMI.