On the Chromatic Uniqueness of Edge-Gluing of Complete Bipartite Graphs and Cycles

In this paper, it is shown that the graph obtained by overlapping the cycle and the complete tripartite graph at an edge is uniquely determined by its chromatic polynomial. ) 3 ( ≥ m Cm 2 , 2 , 2 K Let G be a finite graph with neither loops nor multiple edges and let ) ; ( λ G P denote its chromatic polynomial. Then G is said to be chromatically unique if ) ; ( ) ; ( λ λ G Y P = implies that Y is isomorphic to G. Let n K and denote a complete graph and a cycle respectively on n vertices. The complete t-partite graph whose t partite sets have vertices is denoted by Suppose G and H are two graphs each contains a complete subgraph Let denote any graph obtained by overlapping G and H at In the case that this is sometimes termed as an edge-gluing of G and H. n C t r r r , , , 2 1 . , , , 2 1 t r r r K . n K H G n ∪ . n K , 2 = n Suppose G and H are two chromatically unique graphs. While necessary and sufficient conditions for to be chromatically unique are already known in the literature (see [3], [13], [15] or [10]), not a great deal is known about the chromatic uniqueness of Some necessary conditions were obtained in [3] and [13]. It is asked in [5] (Question 5) whether or not these necessary conditions are also sufficient. No counterexamples are known yet. Some special cases that show that the necessary conditions are also sufficient are given in [2], [4], [6], [8] and [9]. In particular, it is shown in [6] (see also [16]) that is chromatically unique for all and all The more general situation as whether is chromatically unique remains unknown. H G 1 ∪