On Hamiltonian Decomposition Problem of 3-Arc Graphs.

A 4-tuple (, , , ) in a graph is a 3-arc if each of (, , ) and (, , ) is a path. The 3-arc graph of is the graph with vertex set all arcs of and edge set containing all edges joining and whenever (, , , ) is a 3-arc of . A Hamilton cycle is a closed path meeting each vertex of a graph. A graph including a Hamilton cycle is called Hamiltonian and has a Hamiltonian decomposition provided its edge set admits a partition into disjoint Hamilton cycles (possibly with a single perfect matching). The current paper proves that every connected 3-arc graph consists of more than one Hamilton cycle. Since the 3-arc graph of a cubic graph is 4-regular, it further proves that each 3-arc graph of a cubic graph in a certain family has a Hamiltonian decomposition.