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The number of shortest cycles and the chromatic uniqueness of a graph. (English) Zbl 0770.05064
Authors’ abstract: For a graph $$G$$, let $$g(G)$$ and $$\sigma_ g(G)$$ denote, respectively, the girth of $$G$$ and the number of cycles of length $$g(G)$$ in $$G$$. In this paper, we first obtain an upper bound for $$\sigma_ g(G)$$ and determine the structure of a 2-connected graph $$G$$ when $$\sigma_ g(G)$$ attains the bound. These extremal graphs are then more-or-less classified, but one case leads to an unsolved problem. The structural results are finally applied to show that certain families of graphs are chromatically unique.

##### MSC:
 05C35 Extremal problems in graph theory 05C38 Paths and cycles 05C15 Coloring of graphs and hypergraphs
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