Edge and total choosability of near-outerplanar graphs

It is proved that, if G is a K4-minor-free graph with maximum degree ∆ ≥ 4, then G is totally (∆ + 1)-choosable; that is, if every element (vertex or edge) of G is assigned a list of ∆ + 1 colours, then every element can be coloured with a colour from its own list in such a way that every two adjacent or incident elements are coloured with different colours. Together with other known results, this shows that the List-Total-Colouring Conjecture, that ch’’(G) = χ’(G) for every graph G, is true for all K4-minor-free graphs. The List-Edge-Colouring Conjecture is also known to be true for these graphs. As a fairly straightforward consequence, it is proved that both conjectures hold also for all K2,3-minor free graphs and all ( K2 + (K1 U K2))-minor-free graphs

The average degree of a multigraph critical with respect to edge or total choosability

AbstractLet G be a multigraph with maximum degree at most Δ⩾3 such that ch′(G)>Δ or ch″(G)>Δ+1 and G is minimal with this property. A new proof is given for the result (which was already known, apart from a simple calculation) that the average degree of G is greater than 2Δ except possibly in the second case when Δ=5

List-edge and list-total colorings of graphs embedded on hyperbolic surfaces

AbstractIn the paper, we prove that if G is a graph embeddable on a surface of Euler characteristic ε<0 and Δ≥25−24ε+10, then χlist′(G)=Δ and χlist″(G)=Δ+1. This extends a result of Borodin, Kostochka and Woodall [O.V. Borodin, A.V. Kostochka, D.R. Woodall, List-edge and list-total colorings of multigraphs, J. Comb. Theory Series B 71 (1997) 184–204]

Total choosability of planar graphs with maximum degree 4

AbstractLet G be a planar graph with maximum degree 4. It is known that G is 8-totally choosable. It has been recently proved that if G has girth g⩾6, then G is 5-totally choosable. In this note we improve the first result by showing that G is 7-totally choosable and complete the latter one by showing that G is 6-totally choosable if G has girth at least 5

Minimal counterexamples and discharging method

Recently, the author found that there is a common mistake in some papers by using minimal counterexample and discharging method. We first discuss how the mistake is generated, and give a method to fix the mistake. As an illustration, we consider total coloring of planar or toroidal graphs, and show that: if $G$ is a planar or toroidal graph with maximum degree at most $\kappa - 1$, where $\kappa \geq 11$, then the total chromatic number is at most $\kappa$.Comment: 8 pages. Preliminary version, comments are welcom