Thomassen (J. Combin. Theory Ser. B, 62 (1994), pp. 180–181) proved that every planar graph is 5-choosable. This result was generalized by Škrekovski (Discrete Math., 190 (1998), pp. 223–226) and He, Miao, and Shen (Discrete Math., 308 (2008), pp. 4024–4026), who proved that every K5-minor-free graph is 5-choosable. Both proofs rely on the characterization of K5-minorfree graphs due to Wagner (...

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2-regular graph). The List-Colouring Conjecture (LCC) is that every multigraph G has edgechoosability (list chromatic index) ch’(G) equal to its chromatic index x’(G). In this paper the LCC is proved first for multicircuits, and then, building on results of Peterson and Woodall, for any multigraph G in which ...

Thomassen proved that all planar graphs are 5-choosable. Škrekovski strengthened the result by showing K5-minor-free Dvo?ák and Postle pointed out DP-5-colorable. In this note, we first improve these results every or K3,3-minor-free graph is final section, further under term strictly f-degenerate transversal.

A graph G with a list of colors L(v) and weight w(v) for each vertex v is (L,w)-colorable if one can choose a subset of w(v) colors from L(v) for each vertex v, such that adjacent vertices receive disjoint color sets. In this paper, we give necessary and sufficient conditions for a weighted path to be (L,w)-colorable for some list assignments L. Furthermore, we solve the problem of the free-cho...

In this paper, we extend the Grr otzsch Theorem by proving that the clique hypergraph H(G) of every planar graph is 3-colorable. We also extend this result to list colorings by proving that H(G) is 4-choosable for every planar or projective planar graph G. Finally, 4-choosability of H(G) is established for the class of locally planar graphs on arbitrary surfaces.

The (d,1)-total labelling of graphs was introduced by Havet and Yu. In this paper, we consider the list version of (d,1)-total labelling of graphs. Let G be a graph embedded in a surface with Euler characteristic ε whose maximum degree ∆(G) is sufficiently large. We prove that the (d,1)-total choosability C d,1(G) of G is at most ∆(G) + 2d.