Almost Every Complement of a Tadpole Graph is Not Chromatically Unique

Abstract

The study of chromatically unique graphs has been drawing much attention and many results are surveyed in \([4, 12, 13]\). The notion of adjoint polynomials of graphs was first introduced and applied to the study of the chromaticity of the complements of the graphs by Liu \([17]\) (see also \([4]\)). Two invariants for adjoint equivalent graphs that have been employed successfully to determine chromatic unique graphs were introduced by Liu \([17]\) and Dong et al. \([4]\) respectively. In the paper, we shall utilize, among other things, these two invariants to investigate the chromaticity of the complement of the tadpole graphs \(C_n(P_m)\), the graph obtained from a path \(P_m\) and a cycle \(C_n\) by identifying a pendant vertex of the path with a vertex of the cycle. Let \(\bar{G}\) stand for the complement of a graph \(G\). We prove the following results:

1. The graph \(\overline{{{C}_{n-1}(P_2)}}\) is chromatically unique if and only if \(n \neq 5, 7\).
2. Almost every \(\overline{{C_n(P_m)}}\) is not chromatically unique, where \(n \geq 4\) and \(m \geq 2\).