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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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