A graph is pseudo-outerplanar if each of its blocks has an embedding in the plane so that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, the total coloring conjecture is completely confirmed for pseudoouterplanar graphs. In particular, it is proved that the total chromatic number of every pseudo-...

A graph is pseudo-outerplanar if each of its blocks has an embedding in the plane so that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. It is proved that every pseudo-outerplanar graph with maximum degree ∆ ≥ 5 is totally (∆ + 1)-choosable.

Let G be a planar graph with maximum degree Δ. In this paper, it is proved that if Δ ≥ 9, then G is total-(Δ+2)-choosable. Some results on list total coloring of G without cycles of specific lengths are given.

Let G be a 2-connected planar graph with maximum degree ∆ such that G has no cycle of length from 4 to k, where k ≥ 4. Then the total chromatic number of G is ∆+1 if (∆, k) ∈ {(7, 4), (6, 5), (5, 7), (4, 14)}.

A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A {em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has n...