Chromaticity of the complements of paths and cycles Original Research Article

Let Pn and Cn denote the path and cycle of order n. In [6], Guo and Li conjectured that if G is a 2-regular graph without subgraph isomorphic to C4, then the complement of G is chromatically unique. In this paper, we present a proof of this conjecture. We also obtain that if n, is even and ni ≠ 4 mod 10, then the complement of ⋃ki=1Pni is chromatically unique. A new parameter π(G) of graph G and some recursive formulas are introduced as tools, and connected graphs with π(G) = 0 or 1 are characterized. The chromatic uniqueness of certain kinds of graphs is also discussed.