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The Average Degree of Minimally Contraction-Critically 5-Connected Graphs



An edge of a 5-connected graph is said to be 5-removable (resp. 5-contractible) if the removal (resp. the contraction) of the edge results in a 5-connected graph. A 5-connected graph with neither 5-removable edges nor 5-contractible edges is said to be minimally contraction-critically 5-connected. We show the average degree of every minimally contraction-critically 5-connected graph is less than math formula. This bound is sharp in the sense that for any positive real number ε, there is a minimally contraction-critically 5-connected graph whose average degree is greater than math formula.