Total coloring of (sub)cubic Halin graphs

Abstract

Total coloring of a graph is a coloring of its vertices and edges such that adjacent or incident elements receive distinct colors. Total coloring conjecture (stipulating that the total chromatic number of a graph $G$ is at most $Δ(G)+2$) is known to be true for subcubic graphs -- five colors are always enough. However, deciding whether a total coloring with only four colors exists remains a difficult problem, even in the class of bipartite cubic graphs. We solve the problem completely for cubic and subcubic Halin graphs, proving that there are only finitely many such graphs requiring five colors.

Figures (5)

• Figure 1: The only four known cubic Halin graphs with $\chi" = 5$, referred to as $K_4$, $H_4$, $H_5$, and $H_8$ (from left to right).
• Figure 2: A Halin tripole $T$ decomposes into two Halin tripoles $T_1$ and $T_2$.
• Figure 3: The six bad tripoles without unique incompletable palettes.
• Figure 4: $H_3$ -- subcubic Halin graph with $\chi" = 5.$
• Figure 5: A tripole with an empty SND-palette. Assume it has an SND coloring $\varphi$ using only four colors. Without loss of generality, let the colors at $a$ be 0, 1, 2 as in the drawing. Then $\varphi(bd)=3$, otherwise $\varphi[b]\subset \varphi[a]$. To avoid $\varphi[b]\subset \varphi[d]$, the color 2 cannot be used at $d$, and so $\varphi(cd)=0$. But then either $\varphi[c]=\varphi[a]$ or $\varphi[c]=\varphi[d]$, a contradiction.

Theorems & Definitions (15)

• Theorem 1
• Lemma 1
• Lemma 2
• Definition 1
• Lemma 3
• Theorem 2
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7