On Strong Even Cycle Decompositions Of The Line Graphs Of Cubic Graphs

A graph is even-cycle decomposable if its edge set can be partitioned into cycles of even length.We call a graph strongly even-cycle decomposable if the edge set of every subdivision with an even number of edges can be partitioned into cycles of even length.Sometime,the easier way to prove that a graph is even-cycle decomposable is to show that it is strongly even-cycle decomposable.A singed graph G),((50)is called even-cycle decomposable if its edge set can be partitioned into the cycles each of which has an even number of singed edges.It has been proved that a graph G is strongly even-cycle decomposable if and only if the signed graph G),((50)is even-cycle decomposable for any?(50)GE)(such that(50)is even.In 2013,Má?jova and Mazák posed the problem: Let G be a bridgeless cubic graph.Is it true that GL)),(((50)has an even cycle decomposition for any ?(50)GLE))((such that |?| is even? With the help of a computer,they checked that GL)),(((50)of the bridgeless cubic graph G with at most 10 vertices has such property.In the paper,we prove that if G is a cubic graph with order n whose circumference is not less than n-2,then GL)(is strongly even-cycle decomposable.Further,we prove that If G is a bridgeless cubic graph such that G has a cycle C and-CEG)(is a forest consisting of a matching and the caterpillars,then GL)(is strongly even-cycle decomposable.Such cubic graphs closely connect Thomassen's conjecture that every 4-connected line graph is Hamiltonian and Kaiser et al conjecture that every essentially 4-edge-connected graph contains a dominating closed trail.