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 . 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 .