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 pseudo-outerplanar graphs. In particular, it is proved that the total chromatic number of every pseudo-outerplanar graph with maximum degree $\Delta\geq 5$ is $\Delta+1$.
A total coloring of a graph G is an assignment of colors to the vertices and edges of G such that every pair of adjacent/incident elements receive different colors. A k-total coloring of a graph G is a total coloring of G from a set of k colors. The minimum positive integer k for which G has a k-total coloring, denoted by χ (G), is called the total chromatic number of G. It is easy to see that χ (G) ≥ ∆(G) + 1 for any graph G by looking at the color of a vertex with maximum degree and its incident edges. The next step is to look for a Brooks-typed or Vizing-typed upper bound on the total chromatic number in terms of maximum degree. It turns out that the total coloring version of maximum degree upper bound is a difficult problem and has eluded mathematicians for nearly 50 years. The most well-known speculation is the total coloring conjecture, independently raised by Behzad [1] and Vizing [3], which asserts that every graph of maximum degree ∆ admits a (∆ + 2)-total coloring. This conjecture remains open, however, many beautiful results concerning it have been obtained (cf. [5]). In particular, the total chromatic number of all outerplanar graphs has been determined completely by Zhang et al. [7] and that of all series-parallel graphs [4] has been determined completely by Wu and Hu [7].
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. For example, K 2,3 and K 4 are both pseudo-outerplanar graphs. This notion was introduced by Zhang, Liu and Wu in [6], where the edge-decomposition of pseudo-outerplanar graphs into forests with a specified property was studied. In this paper, we prove that the total chromatic number of every pseudo-outerplanar graph with maximum degree ∆ ≥ 5 is exactly ∆ + 1 and thus the total coloring conjecture holds for all pseudoouterplanar graphs.
To begin with, let us review an useful structural property of pseudo-outerplanar graphs which was proved in [6].
Lemma 1. Let G be a pseudo-outerplanar graph with minimum degree at least two. Then (1) G has an edge uv such that d
Here it should be remarked that Lemma 1 is a straightforward simplification of the corresponding theorem in [6] (which had more subcases). Otherwise, G = G -{u, v} admits an (M + 1)-total coloring by induction and every edge of the 4-cycle has at least two available colors since it is incident with at most ∆ -1 colored elements. This implies that one can extend the coloring of G to the four edges ux, vx, uy and vy since every 4-cycle is 2-edge-choosable. At last, the two vertices u and v can be easily colored since they are both of degree two. A(ux), then color vx with c(vx) ∈ A(vx) \ {2} and ux with c(ux) ∈ A(ux) \ {c(vx)}. In each case we have c(vx) c(wy ). Thus we can color vy with c(vy) ∈ A(vy) \ {c(vx)} and wy with c(wy) ∈ A(wy) \ {c(wy )} such that c(vy) c(wy). At this stage, the three vertices u, v and w can be easily colored since they are all of degree two. So we assume that A(ux ) = A(wy ) = {1}. Now we firstly color ux and wy by 1. If 1 A(ux), then color vx with c(vx) ∈ A(vx) and ux with c(ux) ∈ A(ux) \ {c(vx)}. The current extended coloring satisfies that c(vx) c(wy ). Therefore, we can color the remaining elements similarly as before. So we assume that 1 ∈ A(ux). This implies that 1 {c(x), c(xy), c(xx ), c(xy )} and thus we can exchange the colors on ux and xx . By doing so we obtain a new coloring satisfying c(ux ) c(wy ) and therefore we can extend this partial coloring to G by a same argument as above.
Corollary 3. Every pseudo-outerplanar graph with maximum degree ∆ ≥ 5 is (∆ + 1)-total colorable.
Note that every graph with maximum degree ∆ ≤ 5 is (∆ + 2)-total colorable (see [2] and Chapter 4 of [5]). So we also have the following corollary. The graph (a) in Figure 1 is a pseudo-outerplanar graph, since it has a pseudoouterplanar drawing as (b). One can easy to check that it is a graph with maximum degree 3 and total chromatic number 5. Thus the upper bound for ∆ in Corollary 3, although probably not the best possible, cannot be less than 4. At last, we leave the following open problem to end this paper.
Problem 5. To determine the total chromatic number of pseudo-outerplanar graphs with maximum degree four.
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