Truncated degree DP-colourability of $K_{2,4}$-minor free graphs

TL;DR

The paper proves that every 2-connected $K_{2,4}$-minor-free graph that is not a cycle or a complete graph is $5$-truncated degree DP-colourable, thereby answering Hutchinson's question in the affirmative and strengthening the result to DP-colourability. The approach combines a structural decomposition of $K_{2,4}$-minor-free graphs (via outerplanar gadgets and subdividable edge sets) with a DP-colouring framework that uses valid covers and a key coding lemma for two-terminal outerplanar graphs. The main contribution is a full DP-colouring extension across the graph's decomposition, culminating in a 3-step elimination of all possible 3-connected base graphs ( wheels, specific $G$, and twelve small graphs), which proves the desired colourability. This advances understanding of truncated-degree colourability within minor-closed graph families and provides tools applicable to broader DP-colouring problems in outerplanar and related classes. The results have potential implications for list-colouring variants and the design of robust coloring schemes under degree constraints in minor-free graphs.

Abstract

Assume $G$ is a graph and $k$ is a positive integer. Let $f$ from $V(G)$ to $ N$ be defined as $f(v)$ is the minimum of $k$ and $d(v)$. If $G$ is $f$-DP-colourable (respectively, $f$-choosable), then we say $G$ is $k$-truncated degree DP-colourable (respectively, $k$-truncated degree-choosable). Hutchinson proved that 2-connected maximal outerplanar graphs other than the triangle are $5$-truncated degree-choosable, and asked whether the result can be extended to all outerplanar graphs, and the question remained open. This paper proves that 2-connected $K24$-minor free graphs other than cycles and complete graphs are $5$-truncated degree DP-colourable. This not only answers Hutchinson's question in the affirmative, but also extends to a larger family of graphs, and strengthens choosability to DP-colourability.

Figures (5)

• Figure 5: The graphs $G_{8,2,4}$ and $G_{6,2,3}^+$.
• Figure 6: The graphs $A$, $A^+$, $B$, $B^+$, $C$, $C^+$ and $D$.
• Figure 7: Some maximal subdividable sets of $W_4$.
• Figure 8: Maximal subdividable sets of $K_5^-$.
• Figure 9: Maximal subdividable sets of $K_3 \Box K_2$, $K_{3,3}$, $A$ and $D$.

Theorems & Definitions (39)

• Definition 1
• Theorem 1: Hutchinson
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Theorem 2: BKPKO2019
• Theorem 3: ZZZ
• Theorem 4: ZZZ
• Definition 6