On \(\sigma\)-Labeling the Union of Three Cycles

Let \( G \) be a graph of size \( n \) with vertex set \( V(G) \) and edge set \( E(G) \). A \( \sigma \)-labeling of \( G \) is a one-to-one function \( f: V(G) \to \{0, 1, \dots, 2n\} \) such that \( \{|f(u) – f(v)| : \{u, v\} \in E(G)\} = \{1, 2, \dots, n\} \). Such a labeling of \( G \) yields cyclic \( G \)-decompositions of \( K_{2n+1} \) and of \( K_{2n+2} – F \), where \( F \) is a \( 1 \)-factor of \( K_{2n+2} \). It is conjectured that a \( 2 \)-regular graph of size \( n \) has a \( \sigma \)-labeling if and only if \( n \equiv 0 \) or \( 3 \pmod{4} \). We show that this conjecture holds when the graph has at most three components.