Fractional chromatic numbers of cones over graphs (Q2757095)

Let \(G\) be a graph and \(n\) an integer. The \(n\)-cone \(\Delta _{n}(G)\) over \(G\) is the graph obtained by taking the categorical product of \(G\) and a path \(P_{n+1}\) with a loop at one end, and then identifying all the vertices whose second coordinate is the other end of \(P_{n+1}\). When \(n=2\) the resulting cone \(\Delta _{2}(G)\) is Mycielski's construction over \(G\). The main result of this paper is the following formula for the fractional chromatic numbers of all cones over graphs: \(\chi _{f}(\Delta _{n}(G))= \chi _{f}(G)+1/\sum _{k=0}^{n-1}(\chi _{f}(G)-1)^{k}\). This generalizes results given by \textit{M. Larsen, J. Propp} and \textit{D. Ullman} [J. Graph Theory 19, No. 3, 411-416 (1995; Zbl 0830.05027)] for Mycielski's construction (\(n=2\)).