Chia, G. L. Some remarks on the chromatic uniqueness of graphs. (English) Zbl 0672.05033 Ars Comb. 26A, 65-72 (1988). A graph G is quasi-separable if it contains a complete subgraph H such that G-H is disconnected. A quasi-block is a maximal subgraph which is not quasi-separable. “We shall consider the situation in which G consists of exactly 2 quasi- blocks \(Q_ 1\) and \(Q_ 2\). Suppose \(Q_ 1\cap Q_ 2=K_ n\). G is said to have property P if for every \(i=1,2\) there exists \(x_ i\in V(Q_ i)\) such that \(x_ i\) is adjacent to all vertices of \(Q_ 1\cap Q_ 2''\) and to at least one other vertex. Remarks. (3) If G has property P, then G is not chromatically unique (i.e. there exists a graph H not isomorphic to G having the same chromatic polynomial as G). (4) If G is chromatically unique, and if \(Q_ 1\cap Q_ 2\) is a \(K_ 2\), then (ii) \(Q_ 1\) and \(Q_ 2\) are chromatically unique; and (iii) \(Q_ 1\) and \(Q_ 2\) are edge- transitive, and at least one is vertex-transitive. Other results refer to a concept of “overlapping” which is not formally defined. Reviewer: W.G.Brown Cited in 7 Documents MSC: 05C15 Coloring of graphs and hypergraphs 05C40 Connectivity 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) Keywords:chromatic polynomial; chromatic uniqueness; quasi-separable; quasi-block; edge-transitive; vertex-transitive PDFBibTeX XMLCite \textit{G. L. Chia}, Ars Comb. 26A, 65--72 (1988; Zbl 0672.05033)