The size of graphs with clique number m and without nowhere-zero 4-flows

Let G be a 2-edge-connected simple graph on n⩾24 vertices containng no Km+1. Of |E(G)|⩾(n - 17 - k2)+(m - l)(k + 12)+ 33, where ⌊n−17/m⌋, then either G has a nowhere-zero 4-flow, or G can be contracted to the Petersen graph. This is a generalization of a result in [4].