an:06260058
Zbl 1289.05176
Wang, Jianfeng; Huang, Qiong Xiang; Teo, Kee L.; Belardo, Francesco; Liu, Ru Ying; Ye, Cheng Fu
Almost every complement of a tadpole graph is not chromatically unique
EN
Ars Comb. 108, 33-49 (2013).
00329535
2013
j
05C15 05C60
chromatic polynomials; chromatically unique; adjoint polynomials; adjointly unique graphs
Summary: The study of chromatically unique graphs has been drawing much attention and many results are surveyed in [\textit{F. M. Dong} et al., Chromatic polynomials and chromaticity of graphs. Singapore: World Scientific Publishing (2005; Zbl 1070.05038); \textit{K. M. Koh} and \textit{K. L. Teo}, Graphs Comb. 6, No. 3, 259--285 (1990; Zbl 0727.05023); Discrete Math. 172, No. 1--3, 59--78 (1997; Zbl 0879.05031)]. 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 \textit{R. Y. Liu} [Discrete Math. 172, No. 1--3, 85--92 (1997; Zbl 0878.05030)] (see also [Dong et al., loc. cit.]). Two invariants for adjoint equivalent graphs that have been employed successfully to determine chromatic unique graphs were introduced by Liu [loc. cit.] and by Dong et al. [loc. cit.], respectively. In this 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 \(G\) 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 \(m\geq 2\).
Zbl 1070.05038; Zbl 0727.05023; Zbl 0879.05031; Zbl 0878.05030