Small cycle double covers and line graph

A cycle double cover (CDC) of a graph is a collection of cycles of the graph with the property that every edge of the graph is included in exactly two cycles. The Cycle Double Cover Conjecture, proposed by both Seymour and Szekeres, independently, states that every bridgeless graph has a CDC.;A small cycle double cover (SCDC) of a simple graph on n vertices is a CDC with at most n-1 cycles. The Small Cycle Double Cover Conjecture, due to Bondy, states that every simple, bridgeless graph has an SCDC.;If a graph has a CDC with certain properties, then its line graph (a simple graph) has an SCDC. By showing that all complete multipartite graphs have CDCs with appropriate properties, it is thus proved that the line graphs of all complete multipartite graphs except $Ksb{1,2}$ (whose line graph has a bridge) have SCDCs.