Summary: The implicational fragment of the logic of relevant implication, \(R_{\to}\) is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on \(LR_{\to}\), a Gentzen-style calculus. In this paper, we add the truth constant \(\mathbf{t} \text{ to } LR_{\to}\), but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve \(\mathbf{t}\). This calculus, \(LT^{\circ{\mathbf t}_{\to}}\), extends the consecution calculus \(LT_{\to}^{\mathbf{t}}\) formalizing the implicational fragment of ticket entailment. We introduce two other new calculi as alternative formulations of \(R_{\to}^{\mathbf{t}}\). For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of \(R_{\to}^{\mathbf{t}}\). These results serve as a basis for our positive solution to the long open problem of the decidability of \(T_{\to}\), which we present in another paper.