List Edge and List Total Colorings of Planar Graphs without non-induced 7-cycles

The terminology and notation used but undefined in this paper can be found in [1]. Let G be a graph and we use V (G), E(G), F (G), ∆(G) and δ(G) to denote the vertex set, edge set, face set, maximum degree, and minimum degree of G, respectively. Let dG(x) or simply d(x), denote the degree of a vertex (resp. face) x in G. A vertex (resp. face) x is called a k-vertex (resp. k-face), k-vertex (resp. k-face), or k−-vertex, if d(x) = k, d(x) ≥ k, or d(x) ≤ k. We use (d1, d2, · · · , dn) to denote a face f if d1, d2, · · · , dn are the degrees of vertices which are incident with the face f . If u1, u2, · · ·, un are the vertices on the boundary walk of a face f , then we write f = u1u2 · · ·un. Let δ(f) denote the minimal degree of vertices which are incident with f . We use fi(v) to denote the number of i-faces which are incident with v for each v ∈ V (G). Let ni(f) denote the number of i-vertices which are incident with f for each f ∈ F (G). A cycle C of length k is called k-cycle, and if there is at least one edge xy ∈ E(G)\E(C) and x, y ∈ V (C), the cycle C is called non-induced k-cycle.