On Hamiltonian Decomposition Problem of 3-Arc Graphs

A 4-tuple (y, x, v, w) in a graph is a 3-arc if each of (y, x, v) and (x, v, w) is a path. The 3-arc graph of H is the graph with vertex set all arcs of H and edge set containing all edges joining xy and vw whenever (y, x, v, w) is a 3-arc of H. A Hamilton cycle is a closed path meeting each vertex of a graph. A graph H including a Hamilton cycle is called Hamiltonian and H 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.