Even Cycle Decompositions Of The Line Graphs Of 2-connected Cubic Graphs

A decomposition of a graph consists of pairwise edge disjoint subgraphs whose union is the graph.Decomposing the line graphs of cubic graphs into subgraphs with certain properties is a canonical problem in graph theory.Markstr?m conjectured that the line graph of a 2-connected cubic graph admits an even cycle decomposition and proved the conjecture for a 2-connected cubic graph with oddness at most 2.However for 2-connected cubic graphs with oddness 2,Markstr¨om's proof does not cover such graphs without 2-factors consisting of only induced cycles.In the paper,we construct infinitely many 2-connected cubic graphs with oddness 2 and no 2-factors consisting of only induced cycles.Further we prove that the conjecture holds for2-connected cubic graphs with oddness at most 4.