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Chromatic equivalence classes of certain generalized polygon trees. (English) Zbl 0883.05058
The graphs considered in this paper are generalized polygon trees with three interior regions, which can be defined non-recursively as follows: the graph $$G^s_t(a,b;c,d)$$ has at most 4 vertices $$u$$, $$v$$, $$w$$, $$x$$, the are two disjoint paths of lengths $$d\geq 2$$ and $$c\geq d$$ joining $$u$$ and $$v$$, and of lengths $$b\geq 2$$ and $$a\geq b$$ joining $$w$$ and $$x$$, one of length $$s\geq 0$$ joining $$u$$ and $$w$$ (if $$s=0$$ then $$u=w$$) and one of length $$t\geq 0$$ joining $$v$$ and $$x$$ (if $$t=0$$ then $$v=x$$). Let $$C_r(a,b;c,d)$$ be the family $$\{G^s_t(a,b; c,d)\mid r=s+t, s\geq 0, t\geq 0\}$$.
Two graphs are called chromatically equivalent if they have the same chromatic polynomial. Several referenced papers give necessary and sufficient conditions for $$C_r(a,b; c,d)$$ to be a chromatic equivalence class for fixed values of some of the parameters $$r$$, $$a$$, $$b$$, $$c$$, $$d$$, and in [$$(*)$$ Y.-H. Peng, Another family of chromatically unique graphs, Graphs Comb. 11, No. 3, 285-295 (1995; Zbl 0836.05028)] it is shown that $$G^0_1(a,b; c,d)$$ is in a chromatic class by itself if $$\min\{a,b,c,d\}\geq 4$$. The paper under review presents several sufficient conditions for $$C_r(a,b; c,d)$$ to be a chromatic equivalence class, of which the following two are proved: $$\min\{a,b, c,d\}\geq r+3\geq 4$$ (from which the result of $$(*)$$ is derived as a corollary) and $$r\geq a\geq b+3\geq c+ 6\geq d+9$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C05 Trees
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