The use of Euler's formula in (3, 1)*-list-coloring

A graph G is called (k, d)∗-choosable if, for every list assignment L satisfying |L(v)| = k for all v ∈ V (G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. Ko-Wei Lih et al. used the way of discharging to prove that every planar graph without 4-cycles and i-cycles for some i ∈ {5, 6, 7} is (3, 1)∗-choosable. In this paper, we show that if G is 2connected, we may just use Euler’s formula and the graph’s structural properties to prove these results. Furthermore, for 2-connected planar graph G, we use the same way to prove that, if G has no 4-cycles, and the number of 5-cycles contained in G is at most 11+bi≥5 5i−24 4 |Vi|c, then G is (3, 1)∗-choosable; if G has no 5-cycles, and any planar embedding of G does not contain any adjacent 3-faces and adjacent 4-faces, then G is (3, 1)∗-choosable.