A Complete Characterisation of Half-Sunlet Decompositions of Line Graphs of Complete Graphs

Let L(Kn) be the line graph of Kn and let HS6 be the graph obtained from a 4-cycle by attaching a pendant edge at each of two antipodal vertices. We prove that L(Kn) has an HS6-decomposition if and only if n≥4 and n≢3(mod4). The proof splits L(Kn) into smaller line graphs together with graphs W(m,k) built from k triangle bundles glued along shared degree-two vertices, characterises the single-copy case, and handles W(m,2) via a construction attached to each cycle of a cycle decomposition of Km.