Nowhere-zero 3-flows in dihedral Cayley graphs

In this paper we prove that every k  -valent Cayley graph of a dihedral group, where k⩾4k⩾4, admits a nowhere-zero 3-flow.

Research highlights
► We determine that every k  -valent Cayley graph of a dihedral group, where k⩾4k⩾4, admits a nowhere-zero 3-flow.
► We also determine that every loopless Cayley multigraph of a dihedral group and of valence at least 4 admits a nowhere-zero 3-flow.
► We characterize that a Cayley multigraph of a dihedral group admits a nowhere-zero 3-flow if and only if after deleting loops, the resulting graph is of valence 2 or of valence 3 and bipartite or of valence greater than 3.